Israel and Hamas are “on the brink” of agreeing on a cease-fire in Gaza and the release of hostages held there, Secretary of State Antony J. Blinken said on Tuesday, raising hopes for some respite to the violence after more than 15 months of war.

“It’s right on the brink, it’s closer than it’s ever been before,” Mr. Blinken said at an Atlantic Council event in Washington. “But right now as we sit here we await final word from Hamas on its acceptance. And until we get that word, we’ll remain on the brink.”

A person familiar with the negotiations said Tuesday evening that Israel and Hamas had been locked in on the deal, with both appearing prepared to publicly accept it in the very near future. The person, who requested anonymity because a deal had not been announced, said the atmosphere was tense but hopeful.

Neither Israeli nor Hamas officials have publicly confirmed their position on the cease-fire proposal, although Mr. Blinken suggested that Israel was on board with the agreement and that the deal’s fate now rested with Hamas.

Negotiators said Hamas seemed ready to accept the deal, including its details about the exchange of Palestinian prisoners for hostages, and the specific movements of Israeli troops as they withdraw from positions in Gaza, according to the person familiar with the discussions.

But the person cautioned that the agreement has more than 100 separate parts and that no diplomatic deal of that complexity can ever be considered completely done until the parties announce it publicly.

U.S. officials have made optimistic remarks about cease-fire talks in the past. only for negotiations to repeatedly break down into mutual recrimination. And representatives of other mediating countries, including Qatar and Egypt, have warned that even substantial progress could be dashed at the last minute.

“We believe that we are at the final stages, but until we have an announcement — there will be no announcement,” Majed al-Ansari, the spokesman for Qatar’s foreign ministry, told reporters on Tuesday.

Still, in recent weeks officials familiar with the talks have expressed greater hope for a deal. Mediators had “managed to minimize a lot of the disagreements between both parties,” Mr. al-Ansari said, adding that they were focused on “the final details of reaching an agreement.”

Officials in both the Israeli government and Hamas have suggested that they are ready to move forward if the other side signs off. On Monday, a Hamas official said a deal was possible in the coming days as long as Israel did not suddenly change its positions. On Tuesday, an Israeli official said Israel was ready to close the deal and was waiting for Hamas to make a decision.

Some officials have also suggested that a looming deadline had helped close the gap: the end of President Biden’s term and President-elect Donald J. Trump’s inauguration on Jan. 20. That deadline helped negotiators put pressure on both Israel and Hamas to accelerate their decision-making after months of agonizing delay.

Mr. Trump has warned that there will be “all hell to pay” unless the hostages are freed by the time he becomes president. Steve Witkoff, his pick for Middle East envoy, has also made trips to Qatar and Israel, meeting with top officials including Prime Minister Benjamin Netanyahu of Israel.

American officials have said Mr. Witkoff has been an important and supportive player in the cease-fire talks. He has not been involved in the intricate details of the deal, according to the person familiar with the negotiations. But he has made it clear to all of the parties that Mr. Trump wants a deal completed immediately.

They have also said that the momentum for a deal with Hamas accelerated after Israel and the Lebanese armed group Hezbollah reached their own cease-fire deal in late November. That agreement helped isolate Hamas, according to the person familiar with the negotiations, who said discussions became more productive at that point.

If Hamas and Israel conclude an agreement, it would bring some relief to Palestinians in Gaza, who have endured miserable conditions in displacement camps and relentless bombardments by Israel, and for the families of hostages abducted from Israel, who have worried for more than a year about the fate of their loved ones.

“I pray this time the return is real,” said Manar Silmi, 34, a psychologist who hopes to return to the Gaza City home she fled early in the war. “We’ve suffered more than enough.”

In a statement, Hamas said negotiations “had reached their final stages.” The Palestinian armed group’s leadership “hoped that this round of talks would end with a complete and clear agreement,” Hamas said.

Hamas officials negotiating in Doha must obtain the consent of the group’s remaining military commanders inside Gaza for the deal. But communicating with them can be difficult, as they are mostly believed to be in hiding, often leading to delays. They include Mohammad Sinwar, whose brother Yahya led the group before being killed by Israel.

It was still not clear whether Mr. Sinwar had conveyed his position toward the cease-fire proposal to Hamas leaders in Doha.

The framework of the deal was heavily inspired by previous proposals discussed in May and July, said a diplomat familiar with the talks, speaking on condition of anonymity to discuss the volatile negotiations. Those proposals detailed a three-stage cease-fire in which Israeli troops would gradually withdraw from Gaza, as Hamas released hostages in exchange for Palestinians jailed by Israel.

For over a year, international efforts have failed to end the war set off by the Hamas-led attack that killed around 1,200 people in October 2023. Another 250 people were taken hostage to Gaza, according to the Israeli authorities.

In response, Israel launched a military campaign against Hamas that has destroyed much of the enclave and killed at least 45,000 people, according to Gaza health officials, who do not distinguish between civilians and combatants.

Around 105 hostages were released during a weeklong cease-fire in November 2023. The bodies of others were recovered by Israeli troops, and a handful were rescued alive. Roughly 98 hostages are believed to remain in Gaza, around 36 of whom are presumed dead by the Israeli authorities.

During the first phase of the proposed cease-fire — which would last about six weeks — Hamas would release 33 named hostages, most of whom Israel believes are alive, said an Israeli official, who requested anonymity to discuss the sensitive talks. Israel is willing to release hundreds of Palestinian prisoners in exchange, the official said, but the number depends on how many hostages are still alive.

Eli Albag, whose daughter Liri, 19, was abducted from the military base where she served during the Hamas-led attack, met with Mr. Netanyahu on Tuesday evening alongside other relatives of hostages.

Mr. Netanyahu projected optimism, Mr. Albag said. But he said he still found it hard to think about what it might be like to welcome his daughter home.

“We want to see the deal signed first,” he said. “After that, we’ll make room for other thoughts.”

While there is significant public pressure in Israel to reach a deal to free the hostages, many Israelis also fear that a cease-fire would leave Hamas in power in Gaza, allowing it to regroup and plan more attacks.

Two of Mr. Netanyahu’s hard-line coalition allies — Itamar Ben-Gvir and Bezalel Smotrich — have already denounced the proposed agreement as an effective surrender to Hamas. Their two far-right parties could threaten Mr. Netanyahu’s government if they were to withdraw from his ruling coalition over their opposition to a deal.

But were the lawmakers to press a no-confidence vote in Parliament, they would likely struggle to immediately topple Mr. Netanyahu’s government. Opposition parties have broadly committed to propping up Mr. Netanyahu’s coalition, if necessary, to secure the implementation of an agreement that would free the hostages.

In Gaza, Montaser Bahja, a displaced English teacher sheltering in Gaza City, said Palestinians were starting to feel hopeful about a deal after more than a year of hunger and deprivation.

But even if both sides declared a cease-fire, many Gazans were frightened by their uncertain postwar future, Mr. Bahja said. And even if Hamas’s deal secured the release of hundreds of Palestinian prisoners, few would see that as an achievement given the scale of the death and devastation in Gaza, he added.

“Everything is up in the air,” he said. “At this point, people just want it to end.”

Ismaeel Naar and Edward Wong contributed reporting.

The Americans were not happy.

The migrant situation at the border was out of control, they said, and Mexico was not doing enough to stop it, according to officials from both countries.

In fact, the crisis was worse than Mexican officials had been led to believe by their own immigration chief, Francisco Garduño Yáñez.

The revelation in October 2023 led Mexico’s defense secretary at the time to fly into a rage at an emergency meeting, officials with knowledge of the encounter said.

“You fooled me,” the defense secretary, Luis Cresencio Sandoval González, yelled at Mr. Garduño, according to two people familiar with the incident.

The defense secretary regularly briefed Mexico’s then-president, Andrés Manuel López Obrador. But Mr. Sandoval had learned days earlier from the Americans that the migrant crisis was more dire than he realized.

“You hid information from me, making me lie to the president,” the defense secretary lashed out.

It was a tense chapter in U.S.-Mexico relations, according to five Mexican and American officials privy to bilateral talks on migration, and Mr. Garduño, 76, had landed in the middle of it. Beyond being accused of mismanaging and minimizing the migrant crisis, he is separately facing criminal charges in connection with a fire at a migration detention center that killed 40 people in 2023.

Now, as Mexico stands on the precipice of what are expected to be contentious border discussions with the incoming Trump administration, the same Mexican official blamed for mismanaging the migrant crisis, Mr. Garduño, will be a pivotal player in those negotiations. The American president-elect has vowed to begin mass deportations of undocumented immigrants as soon as he takes office.

The Defense Ministry, Mr. Garduño and the agency he led, the National Migration Institute, did not respond to several requests for comment.

Controlling the Mexico-U.S. border is a sprawling endeavor, involving thousands of government agents from both countries. The issue is often used as a political cudgel. U.S. House Republicans accused the Biden administration of failing to control the border and voted to impeach his homeland security secretary, Alejandro Mayorkas.

In Mexico, Mr. Garduño was the one in the cross hairs.

A former director of Mexico’s prison system, he has been criticized for relying on troops to help manage migrant flows. Mr. Garduño’s agency has also been accused of essentially waving migrants through to the northern border for bribes. In interviews, migrants said they had to pay Mexican migration agents to travel through the country to reach the United States.

In 2022, the British Embassy also commissioned a classified report on Mexico’s migration system, a copy of which was obtained by The New York Times. It found systemic corruption in the government’s handling of migrants, including extortion, sexual abuse and collusion with criminal organizations to kidnap migrants for ransom.

In a 2022 interview, Mr. Garduño defended his performance, saying he had fired nearly half of the agency’s employees for extorting migrants. His agency had issued documents to nearly two million migrants from 2018 to 2022, he said, helping to regularize their presence in the country.

It is “a humanitarian policy of integration and brotherhood,” he said.

But interviews with officials from both countries have laid bare the discontent of American officials with how Mexico was handling migration.

In 2023, President Biden’s popularity was slipping ahead of the 2024 elections. Migration was a top concern among American voters. So the president dispatched Secretary of State Antony J. Blinken and Mr. Mayorkas for an emergency meeting in Mexico City that October.

They told Mr. López Obrador that American border agents had encountered nearly 220,000 migrants at the southern U.S. border that September — one of the largest flows ever recorded, officials with knowledge of the meeting said.

Border patrol agents were overwhelmed. The freight trains from Mexico to the United States had no security. Corrupt conductors, the Americans said, were stopping or slowing the trains to allow migrants to hop on.

They asked Mexican officials to move more aggressively to break up large groups of migrants heading to the U.S. border and to end visa-free travel for countries whose nationals used Mexico to enter America illegally, officials said.

The reality that the American delegation revealed was grimmer than the one presented by Mr. Garduño’s agency, which gave daily briefings to the Mexican administration on the number of migrants intercepted in southern Mexico.

Three officials working on migration and were privy to those figures said the numbers rarely correlated with the data presented by the U.S. Customs and Border Protection and the government of Panama, which many migrants pass through to reach Mexico.

The Mexican military reported that it and the migration agency encountered five million migrants from 2018 to 2024, but Mexico’s Interior Ministry reported about half that number in that time. The 2023 numbers varied widely as well; the migration agency reported nearly 1.5 million encounters that year, whereas the Interior Ministry reported about 500,000.

“Mexico’s government is blurring the picture by issuing two widely divergent numbers, without even explaining the divergence,” said Adam Isacson, a director at the Washington Office on Latin America, a research institute. “It’s confusing, it undermines the government’s credibility, and it makes it harder to anticipate emerging trends.”

After the U.S. delegation returned to Washington, Mr. López Obrador called the emergency meeting of Mexico’s most senior security and migration officials on Oct. 13, 2023. It was held in Tapachula, a city on the border with Guatemala and a funnel for migrants entering Mexico.

The city’s refugee agency was about to collapse, with about 7,000 migrants a day flooding its offices to register as asylum seekers — a fast track to receiving a migrant permit.

The permits were sort of a golden ticket: They allow asylum seekers to study, work and get access to basic services. Though asylum seekers are supposed to stay in the state where they apply, many use the Mexican permits to navigate to the U.S. border without being detained, officials say.

At the emergency meeting, the interior secretary at the time, Luisa María Alcalde Luján, zeroed in on the permits, officials said.

She grilled Mr. Garduño about whether his agency was handing out the permits but allowing asylum seekers to head north toward the U.S. border, according to four officials with knowledge of the meeting, two in attendance.

Yes, Mr. Garduño replied.

As Ms. Alcalde berated him, Mr. Garduño looked down at his lap and fell silent, officials with knowledge of the encounter said.

She then announced to the room that she was stripping Mr. Garduño of the ability to hand out new migration permits without the approval of other government branches.

Ms. Alcalde did not respond to requests for comment.

As soon as the migrant permits stopped, thousands of asylum seekers in Mexico were plunged into legal limbo.

The move made them “easier prey for criminal groups,” said Dana Graber Ladek, the Mexico chief of mission for the International Organization for Migration. It left “migrants with basically no option to be able to work legally in the country,” she added.

Eventually, Mexico restarted issuing the migration permits, but today they are a trickle of what they once were: Only about 3,500 permits were issued last year, compared with nearly 130,000 in 2023.

After the meeting, Mr. Garduño quickly moved to demonstrate that his agency was capable of controlling migrant flows, officials said.

His agents made it harder for migrants to reach the U.S. border and stepped up security on the trains many used to travel north. The number of migrants encountered at the U.S.-Mexican border dropped from September to November by nearly 13 percent, according to November 2023 statistics from the U.S. Customs and Border Protection.

But just as the numbers trended down, a leak prompted high-level officials to call another emergency migration meeting in Mexico.

The Mexican treasury secretary temporarily stopped funding parts of the government in November 2023, including Mr. Garduño’s agency, because of budgetary constraints. But instead of lobbying the treasury to release funds, as other officials did, Mr. Garduño proactively halted his agency’s operations.

On Dec. 1, he sent a memo ordering his agency to pause deportation flights that carry undocumented migrants, withdraw personnel from checkpoints and shut down the busing program that had relieved pressure on the northern border.

The memo was swiftly leaked and went public.

Migrants rushed to the U.S. border, many unhindered by Mexican migration agents. That December, U.S. Customs and Border Protection registered the highest number of migrant encounters on the border in history: nearly 250,000 migrants.

Overwhelmed American border patrol agents shut land border crossings in Lukeville, Ariz., and San Diego. The U.S. border protection agency suspended several railway crossings in Texas.

Mexico’s government, trying to contain the fallout, publicly pledged more funds to its migration agency. Mr. Blinken flew back to Mexico City, on Dec. 27 — with an even larger delegation.

The next month, January 2024, after Mexico and the United States cooperated to enforce stricter measures, the migrant flow at the U.S. border was cut in half.

The pressure from Washington has continued to work; unlawful border crossings have declined. Last June, Mr. Biden issued an executive order to essentially block undocumented migrants from receiving asylum at the border.

Mexico has deployed National Guard troops to immigration checkpoints and bused migrants farther south, exhausting their efforts to head north. The authorities have also broken up migrant caravans so they no longer reach the U.S. border.

In October, Claudia Sheinbaum was sworn in as president of Mexico. She named a new immigration chief, but said Mr. Garduño would continue to advise the government to create a “profound transformation” of its migration agency and to help weather the storm after Mr. Trump takes office Jan. 20.

Mr. Garduño still faces criminal proceedings over the migration center fire. Several Mexican and American officials said they thought he would resign after the tragedy. But he has been a confidante of Mr. López Obrador for decades.

Mr. Garduño is not under arrest, but every two weeks, he must check in with the prosecuting judge.

Emiliano Rodríguez Mega and Paulina Villegas contributed reporting.

Ukraine carried out “massive” drone strikes on several regions of Russia overnight, local officials there said on Tuesday, in what appeared to be one of the largest recent assaults in Kyiv’s campaign to cripple Russia’s war machine on its home turf.

The attacks, mostly in southwestern Russia, were the latest in a series demonstrating Ukraine’s ability to strike deep inside the country, even as Kyiv’s forces face setbacks on their own territory.

Blasts were reported in the border region of Bryansk, and drones targeted regions well beyond it like Saratov and Tula in western Russia, officials in those areas said.

Russia’s Defense Ministry said on Tuesday that Ukraine had launched more than 140 drones, along with U.S.-made long-range missiles known as ATACMS and Storm Shadow missiles provided by Britain.

“These actions of the Kyiv regime, supported by Western curators, will not go unanswered,” the ministry said in a statement.

The Ukrainian military’s General Staff said it had attacked three regions of Russia and the republic of Tatarstan, including strikes nearly 700 miles into Russia. The targets included an oil storage base, as well as military production facilities, the General Staff said in a statement.

The drone strikes forced at least six cities to restrict their airspace on Tuesday morning, according to a statement from Russia’s Federal Aviation Agency. Those included the cities of Saratov and Engels, which were attacked overnight.

Two industrial plants sustained damage, Roman V. Busargin, the governor of the Saratov region, wrote on the Telegram messaging app. “Today Saratov and Engels were subjected to a massive UAV attack,” he said, using the initials for unmanned aerial vehicle. “Air defenses eliminated a large number of targets.”

It was the second time in a week that Engels, which is the site of an airfield for some of Russia’s long-range, nuclear-capable bombers, was attacked. Emergency crews recently extinguished a large fire sparked by a strike on Jan. 8.

Mr. Busargin said that, in the wake of the attacks, schools would be closed and classes in Engels and Saratov would be held remotely on Tuesday.

The authorities in the Tula region of western Russia similarly confirmed a large drone attack. Dmitry V. Milyaev, the regional governor, said air defenses had shot down 16 drones and that falling debris had damaged some cars and buildings. There were no casualties, officials said.

Local news media in Kazan, the capital of the republic of Tatarstan, in southwestern Russia, reported that a tanker at a liquefied natural gas base had been struck, igniting a large fire.

The head of the republic, Rustam N. Minnikhanov, wrote on Telegram that firefighters had put out the blaze and that there had been no casualties or “significant damage.”

In the border region of Bryansk, which has come under more regular attack, powerful explosions were reported. The Russian independent news organization Astra said that a chemical plant had been struck; the report could not be independently verified.

Russia’s Defense Ministry said it had shot down 31 drones, six ATACMS and six Storm Shadow missiles targeting Bryansk.

From the start, Kyiv’s strikes in Russia have aimed at limiting Moscow’s ability to attack Ukrainian cities. But in recent weeks, they have taken on added weight as Kyiv has apparently attempted to project strength before President-elect Donald J. Trump’s inauguration next week, amid concerns he might temper U.S. support for Ukraine.

European leaders have assured Ukraine that they will not abandon it in the face of continued Russian aggression.

On Tuesday, Germany’s defense minister, Boris Pistorius, made an unannounced visit to Kyiv to meet with President Volodymyr Zelensky in a show of solidarity.

“It is important for me to show with this trip that we continue to actively support Ukraine,” Mr. Pistorius said upon arriving in the Ukrainian capital, according to German news media.

Mr. Zelensky said the two men had discussed a range of topics, including military assistance to Ukraine, strengthening air defenses and ammunition supplies.

“I am grateful to Germany for its unwavering support, which has not only strengthened Ukraine’s defense, but also played a crucial role in enhancing Europe’s security as a whole,” Mr. Zelensky said in a statement.

Mr. Trump has vowed to end the war swiftly. While he has not said how, many in Ukraine fear that he could make concessions to President Vladimir V. Putin of Russia that have been red lines for Kyiv.

Ahead of Mr. Trump’s inauguration on Monday, the Russian military has also been putting on a show of force. While Kyiv’s drones were targeting regions of Russia overnight, Moscow’s forces were carrying out an aerial assault on Ukraine that put much of the country under air-raid alerts.

Ukraine’s Air Force said on Tuesday morning that nearly 80 drones were involved in the attack but that it had managed to shoot down 60. Apartment buildings and cars sustained damage from downed drones in several regions, it added, but there were no casualties.

Nataliia Novosolova and Nataliya Vasilyeva contributed reporting.

Criminal investigators armed with a court-issued warrant began a second, much-anticipated operation early Wednesday to detain the impeached president, Yoon Suk Yeol, for questioning in connection with insurrection charges that stem from his short-lived imposition of martial law last month.

Police buses started massing before dawn outside the hilltop presidential compound where Mr. Yoon has been holed up since he was impeached — and suspended from office — by the National Assembly on Dec. 14. He was the first South Korean leader to place his country under military rule since the country began democratizing in the late 1980s.

Investigators and police officers gathered at the main gate of Mr. Yoon’s residence before dawn broke on Wednesday, but faced a barricade of buses and presidential security guards blocking the gate leading up to Mr. Yoon’s residence. The standoff was a repeat of when the investigators first visited his residence to serve a detention warrant on Jan. 3. Then, they were outnumbered by presidential security agents and had to beat an embarrassing retreat after a standoff that lasted five and a half hours.

The investigators have regrouped since their initial failed attempt to detain Mr. Yoon, renewing their warrant and saying they would bring in more police officers the next time they visited his residence. Police officials said they planned to deploy 1,000 officers, including units specializing in targeting drug and organized crime gangs, and vowed to arrest any people who obstructed their way.

On Wednesday morning, with Mr. Yoon’s lawyers, lawmakers from his party and personnel from the presidential security service standing outside the compound gates, it appeared that he and his supporters were gearing up to resist the renewed effort for his detention.

Live footage of the street leading up to his compound in the morning showed a tense standoff in below-freezing temperatures, with some shoving and physical struggles at one point.

Since the first attempt to detain Mr. Yoon his security guards have fortified the compound by deploying more buses and razor wire to block gates and walls. Mr. Yoon has vowed to “fight to the end” to return to office and said he won’t surrender to a court warrant that he considered illegal.

South Korea’s acting president, Deputy Prime Minister Choi Sang-mok, warned government agencies involved in the standoff against violence.

“All the people and the international community are watching this,” he said in a statement. “We cannot tolerate physical violence for any purposes because it will irreparably damage the trust of the people and our international reputation.”

The effort to take in Mr. Yoon is the first time in South Korean history that the authorities are trying to detain a sitting president. The unfolding events have gripped the country, with news and social media channels livestreaming coverage. There are fears of a violent clash if neither side backs down.

A day before, the Constitutional Court began a hearing on whether to unseat Mr. Yoon, who did not show up. His lawyers said he feared the investigators would detain him if he left his presidential compound.

During the last attempt to serve the warrant, the Presidential Security Service, a government agency assigned to protect the president and his family, outnumbered the Corruption Investigation Office for High-ranking Officials, or C.I.O., which sought to detain him with the help of the police. It deployed 200 bodyguards and soldiers to block 100 C.I.O. agents and police officers.

It was unclear how much manpower the presidential security team could muster in its efforts to stop the new attempt to detain Mr. Yoon.

A military unit guards the perimeter of the presidential compound. But after the first standoff, the Defense Ministry told the presidential security service that it could no longer use the soldiers to block the serving of the warrant, saying this was not part of the soldiers’ duty.

He is a deeply unpopular figure in Britain, according to opinion polls, and his social media channel has lost users in the country since he took it over in October 2022. Yet when Elon Musk put Britain in his cross hairs on X in recent weeks, pounding the political establishment over a decade-old child sex abuse scandal, he instantly catapulted the issue to the top of the news agenda.

Mr. Musk’s success is rooted in two obvious factors: his mammoth fortune and his alliance with the incoming president of the United States, Donald J. Trump. But it also reflects a British political and media establishment that is divided and deeply in flux, all of which has made Britain easy pickings for an outside influencer with vast resources and a single-minded mission to disrupt.

Britain’s right-leaning newspapers have picked up and amplified Mr. Musk’s call for a new national investigation of young girls who were sexually exploited in several towns in the 2000s, including in Rotherham, where an estimated 1,400 girls were exploited by “grooming gangs” composed largely of British Pakistani men.

The leader of the opposition Conservative Party, Kemi Badenoch, threw her support behind a new national investigation as well, even though the previous Conservative government did not pursue one. A former Tory leader, Boris Johnson, once belittled those inquiries, saying, an “awful lot of money and an awful lot of police time now goes into these historic offenses and all this malarkey.”

The Labour government, which was vaulted into power with a landslide majority in July, has so far rejected Tory calls for another investigation, saying its priority is to implement the recommendations from a previous seven-year-long national investigation, including tightening requirements to report child abuse and collect better data on cases.

The government’s minister for safeguarding and violence against women and girls, Jess Phillips, had earlier pushed back privately on calls for a fresh national inquiry, arguing that a local inquiry would be more effective, for which she was labeled a “rape genocide apologist” in a post by Mr. Musk.

But even as Prime Minister Keir Starmer defended Ms. Phillips, the drumbeat of angry Musk posts forced him to retreat from his claim that Ms. Badenoch and others were calling for an investigation simply “because they want to jump on a bandwagon of the far right.” Now, he says, he has an open mind about it.

The crimes committed against young girls in these cases were so appalling that the resurfacing of them, even years later, would have drawn a strong public reaction, regardless of how they came to light.

But as Sunder Katwala, the director of British Future, a research organization in London, said, “Elon Musk’s political posting has been enormously influential in driving both the political and media conversation in the U.K.”

Based strictly on the numbers, Mr. Katwala said, that makes little sense: Mr. Musk is viewed favorably by only 20 percent of people in Britain, according to the market research firm YouGov, while 71 percent view him unfavorably. The reach of X fell by eight percent from May 2023 to May 2024, according to an annual report on online usage by Ofcom, which regulates communications in Britain.

“There has been a recent tradition of lecturing the liberal left to remember that Twitter is not Britain,” Mr. Katwala said. “But the right-wing ecosystem seems inclined to forget that X is not Britain either.”

Still, he said, X has managed to hang on to its followers among the politicians and journalists who work in Westminster and who set the political news agenda. They have proven a rapt audience for Mr. Musk’s increasingly strident, often erroneous, posts about Britain and his false claims about Mr. Starmer’s supposed complicity in a scheme to cover up child sex abuse, dating back to his days as Britain’s director of prosecutions. In fact Mr. Starmer’s office brought the first case against a grooming gang and drafted new guidelines for the mandatory reporting of child sex offenses.

While many of Mr. Musk’s posts, particularly those on grooming gangs, originated in the ecosystem of far-right bloggers and activists, they are also tempting to mainstream politicians in search of a cudgel to use against their opponents. And they appeal to editors and broadcasters looking for a good story.

“The British press and the broadcasters, to a degree, fell all over themselves to give Elon Musk publicity,” said David Yelland, a former editor of Rupert Murdoch’s tabloid, The Sun. “In the print press, they did it because they are extremely hostile to Keir Starmer. This is plain old Fleet Street bias.”

Claire Enders, a London-based media researcher and founder of Enders Analysis, likened Mr. Musk to Mr. Murdoch, the insurgent media baron from Australia who upended the London newspaper industry in the 1970s. “We just have a new Murdoch,” she said. “He’s American, he’s a multibillionaire, and he’s close to Trump.”

Mr. Musk, however, is not interested in taking over the British press so much as discrediting it. He claims the news media was complicit in a coverup of abuses against young girls. The truth is, British newspapers across the political spectrum did cover these crimes, if not immediately, then energetically, as the scale of the abuses became apparent in the late 2000s and early 2010s. The Times of London published a major investigation of the scandal, and the slow response to it by the police, in 2011.

“It’s been on the front page of every paper and led the 6 o’clock news for years,” said Raheem Kassam, who covered the scandal as editor of the British outpost of the right-wing news outlet, Breitbart News. “The idea that there is a media blackout on this, and we needed Elon Musk to uncover it, is nonsense.”

And yet even The Times of London, which is also owned by Mr. Murdoch, published its own demand for a new inquiry on its front page on Jan. 6. It pointed to the hundreds of articles it had published on the scandal but added that a newspaper “can only go so far.”

No British media outlet has revived the grooming scandal with the zeal of GB News, a hard-right cable news channel that went on the air in 2021, a decade after The Times’s investigation into grooming gangs. Charlie Peters, an investigative reporter, broke the story that Ms. Phillips had rejected a request for a national inquiry into child sexual abuse in Oldham, a town near Manchester.

Since then, GB News has interviewed relatives of the victims of abuse across Britain. Nigel Farage, the leader of Reform U.K., an anti-immigrant party, has praised Mr. Peters, saying he had “really reignited this story” and demonstrated that “these barbarities have taken place in at least 50 towns.” Mr. Farage has vowed that Reform will carry out its own investigation if the government does not.

But Mr. Farage is also trying to extricate himself from his own spat with Mr. Musk — this one over Mr. Farage’s refusal to echo Mr. Musk’s demand that Tommy Robinson, a far-right agitator with multiple criminal convictions, be released from prison. Mr. Musk posted that Reform needed to find a new leader because Mr. Farage “doesn’t have what it takes.”

The cumulative effect of Mr. Musk’s inflammatory posts has been to energize Britain’s populist right. Even Mr. Farage’s rift with Mr. Musk may ultimately play to his benefit, giving him credibility with those who revile Mr. Robinson. The Labour government, meanwhile, is struggling to regain control over its agenda, while critics say the Conservatives embraced Mr. Musk largely to stay relevant.

“He brings the two tactical nuclear weapons of modern politics: unlimited cash and a social media platform,” said Stephen K. Bannon, a former adviser to Mr. Trump who has clashed with Mr. Musk on visas for skilled workers in the United States. “There’s not a government in Europe that can withstand this guy’s onslaught.”

Mr. Kassam, a former chief of staff to Mr. Farage who now works with Mr. Bannon in the United States, said he expected Mr. Farage and Mr. Musk to mend fences soon. Aside from their differences over Mr. Robinson, he said, their political philosophies were probably closer than those of Mr. Musk and Mr. Trump.

“Elon tries to break things and see where they fall,” Mr. Kassam said. “He’s treating the United Kingdom like one of his companies.”

The teacher needed teenagers for her summer acting class in Kyiv, which would end with the performance of an original play.

“This is a course for happy children, free in their thoughts and dreams,” the instructor, Olesia Korzhenevska, wrote on Facebook last spring.

It was hard to find happy teenagers in Ukraine. The pandemic and the war with Russia had trapped some young people in their homes, solitary and fearful, for more than four years. Many did not know how to socialize and could not imagine a future without war.

But two days after her Facebook post, Ms. Korzhenevska heard from the mother of a 16-year-old boy, asking her to accept him in the class.

Sasha Suchyk was an unlikely candidate. A year earlier, he had dropped out of the same class and landed in a mental hospital, suffering from clinical depression, even hurting himself. Buffeted by the war and dark thoughts, he was still in the hospital, where he had spent most of the previous year.

“Your lessons for him would be about the opportunity to open himself up and find new friends,” his mother, Olena Suchyk, told the teacher.

Ms. Korzhenevska, 40, remembered Sasha. Skinny, with long brown hair and a somewhat vacant look. He had disappeared after only a few classes. But now he sent her a video of himself, and she saw he had gained weight. His hair was short. He smiled.

“I’ve been playing guitar for four years and played violin for five years,” Sasha said. “I want to join the course to develop my creative potential and make new friends.”

Ms. Korzhenevska was not trained to work with troubled teenagers. But she was a patient teacher, and she had learned a lot raising her own teenage son, who was autistic.

“This is quite a challenge,” she remembered thinking about Sasha. “But I accept it.”

Sasha got out of the hospital in June. For the next three months, he and three other young actors tried to put aside their worries and work on the play Ms. Korzhenevska wrote for them. Its theme was that life could work out even if everything seemed to be falling apart.

The title of the play was “It’s okay!” But could it be, really?

The Teacher

Ms. Korzhenevska had worked as an event planner, teacher and film producer before she started teaching acting classes to teenagers during the pandemic.

A building in Kyiv’s hipster neighborhood of Podil was her creative laboratory. With its brick walls painted white, hardwood floors and high ceilings, the ground floor vaguely resembled a tech entrepreneur’s Manhattan loft. Ms. Korzhenevska named it the 9¾ School, after the magical train platform in the Harry Potter books, and offered classes mainly on the weekends.

After Russia invaded Ukraine in 2022, Ms. Korzhenevska used the space to also teach military recruits to operate drones and run drills. Upstairs, teachers worked with her son and another autistic teenager.

Ms. Korzhenevska wrote a new play for every acting class. After the invasion, she focused on war stories because many students had loved ones fighting near the front lines. In 2023, the students got “Turtle in the Pot,” so named because one teenager’s family had fled their home carrying their pet turtle in a pot.

Ms. Korzhenevska noticed right away that the vibe in 2024 was different. Everyone needed a break from the war. She wanted to help the students imagine themselves in a more predictable, more routine environment. Someplace like America, Ms. Korzhenevska thought, where none of them had ever been.

She needed a break, too. Her fiancé, Dani, whom she had met at a music festival in 2017, had joined the army the day after the Russians invaded, and he was still on the eastern front, flying drones.

The Play

When creating her plays, Ms. Korzhenevska looked to the students for inspiration.

The 2024 class had four students. Solomia Cherepushko-Zagrebelna, a 13-year-old who goes by Solya, spent hours a day on her beauty ritual — maintaining stiletto nails and eyelashes that looked like awnings. But in class, she was serious, the student most interested in the craft of acting.

Anna Yuzhda, 14, wore glasses and seemed nerdy, but she played the guitar and exuded cool. Ms. Korzhenevska decided they could be sisters, one beautiful and one brainy.

A third student, Alisa Pazushko, was an old soul at age 12. Two years earlier, as the Russians besieged her home of Mariupol, her mother woke her one morning and told her to pack. She grabbed two books — “How to Train Your Dragon” and a Harry Potter — but left behind her favorite stuffed animal, a gray-and-black cat, and with her family, fled to a new life in Kyiv.

Alisa attended online classes from Kyiv, and so had not made friends in her new city. Tall for her age, she seemed as if she could use something to care for, Ms. Korzhenevska thought. Alisa could play the mother in the story that was beginning to take shape in Ms. Korzhenevska’s head.

The rough outline: A teenage boy from an affluent New York City family was orphaned in a car accident and sent to live in rural Mississippi with his mother’s best friend, who was so poor she could not even afford pancake syrup. The woman had two daughters: a smart bookworm and a beautiful cheerleader. The boy, Simon, fell in love with both.

Sasha would play Simon.

Ms. Korzhenevska picked her setting after meeting an American in a Kyiv bar who extolled the virtues of his hometown: West Point, Miss., a city of 10,000 with a website boasting that it “embodies what was best about America a generation ago.”

She included two American songs. One was “Oceans (Where Feet May Fail),” by Hillsong United, a reminder to keep faith in God, even when things seemed difficult. The other was performed by Jane Marczewski, known as Nightbirde, who became an international sensation after singing it on “America’s Got Talent” when she had terminal cancer.

That song, “It’s OK,” gave the play its title. Ms. Korzhenevska would say later that she wrote it with Sasha in mind.

The Star

On a Sunday in July, a generator sat near the front door of the theater in case the electricity went out, as it often did when Russia attacked Ukraine’s power supply. Air-raid sirens punctuated the hum of traffic. It was about 90 degrees.

But on the makeshift stage, it was Mississippi. Sasha, playing Simon, slumped into the room and flopped glumly onto a chair. Too sad, Ms. Korzhenevska thought. By this point in the script, Simon had been living with his new family for a few months.

“You’re still sad, but a little more fun,” Ms. Korzhenevska explained. “You’ve been here for a while, and so you’re a little more cheerful. You were terrible once, but not so much anymore. You can smile now.”

Sasha tried it again, with a hint of a smile. Angst with possibility, a singular teenage emotion.

The pandemic had been hard for Sasha, who had gone to school online and spent a lot of time alone. Once the war began, his mother and stepfather sent him to Poland, where he would be safer, to live with his father.

For almost a year, Sasha bounced between his parents, depending on whether his school in Kyiv was open. In the chaos, the sadness that put him in the hospital took over.

The cast did not talk about such things. They focused on the project.

Just as Sasha had the central role in the play, he became the center of the class, with the three younger girls seeming to fawn over him. With Anna, he practiced Nirvana songs from the play on the guitar. Alisa preferred talking to Sasha over anyone else.

“We have more interests in common than with the other girls,” Alisa said.

The students learned as they went. Ms. Korzhenevska taught Sasha how to hold his skateboard in the middle, so it did not hang awkwardly. She told Anna, who played the brainy sister, that she needed to hand an apple to Sasha in a way that conveyed flirtation. The young actors worked hard, memorizing their lines. Sasha learned a poem about loss and hope.

“And even if your soul is the most desolate of deserts, then something will grow from it,” he repeated.

Still, the war intruded. Ms. Korzhenevska saw a psychiatrist to cope with her worry about her fiancé and her country, but the medication made her want to sleep all the time. On some days, she couldn’t get out of bed.

“The only thing managing to get me out of my house is this play,” she said. “For the rehearsal, I am fine.”

Dani — whose full name is not being published because of military rules — was in charge of a group of drone operators near the eastern town of Pokrovsk. On Sept. 6, a car carrying two of his soldiers hit a land mine. The soldier driving lost the lower part of her left leg. Dani sent a video to Ms. Korzhenevska of the panicked trip to evacuate her, and they cried together while watching it.

Nine days later, the play would premiere.

No Regrets

Outside the theater, more than 40 people, including Sasha’s mother, waited, dressed in Sunday outfits and holding bouquets. Some had not been to the theater in years.

Inside, Sasha sat on the dressing room floor in shorts and his favorite shirt, which had English words like “rebel” printed on it. He chewed the inside of his lip. His face, always expressive, settled somewhere between startled and amused.

Alisa paced. Sasha and the two other girls tried relaxation techniques: shaking out their hands, playing meditation music. Would they be able to avoid laughing when they sang American songs?

Ms. Korzhenevska introduced the production, wearing a blue and white polka-dot dress and with her blond hair pulled back.

“We are in the middle of a war,” she told them. “We have been talking about war for a long time. But this performance is different. We wanted to show something easy, romantic and not about war.”

Alisa came out first. Soon, Sasha appeared as Simon. Ms. Suchyk, overwhelmed to see him in such a prominent role, began to cry.

Sasha forgot a line, as did one of the girls. In the audience, no one knew. As the story unfolded, Simon fell for both sisters and began to accept his parents’ death. In the end, he moved on but left gifts: pancake syrup, a sparkling dress designed by his mother, who had been a fashion designer, and $2,000 so the brainy girl could get Lasik eye surgery.

The audience responded as if the play had released something in them that they had been holding back. “Nobody died in the end and everything was OK,” Ms. Korzhenevska said. “But people were crying.”

Alisa’s mother said no one should judge the performance by her family’s reaction, as they all had post-traumatic stress disorder. Tears streamed down the face of Alisa’s aunt, whose former husband disappeared and was presumed dead after Russian troops took over Mariupol.

Sasha said the class had helped him make friends and return to school. He now wants to become a psychologist, he said, to help military veterans and teenagers.

He talked about his character, Simon, as if he were real.

“I know Simon is pretty sad but with that family that loves him, the character, he got loved by someone,” Sasha said. “It was very good for him.”

After the performance, Ms. Korzhenevska joined the actors onstage and praised each one. Sasha, she said, had developed a kind of peace and inner calm.

“I’m just on tranquilizers,” Sasha said. The audience laughed.

“Me too,” Ms. Korzhenevska admitted.

“I’m just joking,” he replied.

Ms. Korzhenevska hugged him. “I’m not,” she said.

Evelina Riabenko contributed reporting.

Audio produced by Sarah Diamond.

Prime Minister Keir Starmer of Britain suffered a new blow on Tuesday when his anticorruption minister, Tulip Siddiq, quit her post weeks after being named in an embezzlement investigation in Bangladesh.

Ms. Siddiq, 42, is the niece of Sheikh Hasina, the former prime minister of Bangladesh, who resigned last year after 15 years in power and fled the country amid a broad student-led protest movement against her repressive rule.

A junior minister in Mr. Starmer’s government, Ms. Siddiq had previously referred herself to the prime minister’s ethics adviser for investigation after questions arose over whether she had benefited financially from her ties to Ms. Hasina.

Ms. Siddiq has dismissed the allegations against her as politically motivated and insisted that she did nothing wrong. But in an official letter of resignation to Mr. Starmer on Tuesday, she wrote that the media focus on her risked diverting attention from the government’s political agenda.

“I want to assure you that I acted and have continued to act with full transparency and on the advice of officials on these matters,” Ms. Siddiq wrote. “However it is clear that my continuing in my role as economic secretary to the Treasury is likely to be a distraction from the work of the government.”

As economic secretary to the Treasury, a position she was given when the Labour Party came to power last July, Ms. Siddiq was responsible for tackling corruption in financial markets, including money laundering and illicit finance.

Mr. Starmer’s ethics adviser, Laurie Magnus, said on Tuesday that having reviewed the facts regarding Ms. Siddiq’s case, he concluded that she had not broken the code under which ministers serve. In a letter to the prime minister, he said he had found no evidence of impropriety or that she had made unusual financial arrangements.

However he added that it was “regrettable that she was not more alert to the potential reputational risks — both to her and the government — arising from her close family’s association with Bangladesh.”

And he added, “Unfortunately, I have not been able to obtain comprehensive comfort in relation to all the UK property-related matters referred to in the media.”

The departure is the second high-profile resignation from Mr. Starmer’s government in recent months, after Louise Haigh quit as transport secretary in November when it emerged that she had been convicted of a fraud offense involving a phone a decade ago.

Unlike Ms. Haigh, Ms. Siddiq was not a member of the cabinet, but she was regarded as a political ally of Mr. Starmer’s, and the nature of her government role added to the embarrassment. She has been replaced by Emma Reynolds, another junior minister.

Ms. Siddiq was named in December in an investigation into claims that her family had embezzled up to £3.9 billion pounds, or nearly $5 billion, from infrastructure projects in Bangladesh.

Officials of the new Bangladeshi government have accused Ms. Hasina and her associates of siphoning off billions of dollars from the country every year, bringing its economy to the verge of collapse.

Ms. Siddiq was born in London. Her father was an academic, and her mother, Sheikh Rehana, is the sister of Ms. Hasina. The two sisters were the only survivors of a 1975 military coup that massacred the rest of their family, including their father, Sheikh Mujib ur Rehman, the founding leader of Bangladesh.

The furor over the Bangladeshi investigation prompted a wider focus in the British news media on Ms. Siddiq’s ties to her aunt, including reports that she had lived in London properties with links to Ms. Hasina.

Bangladesh’s interim leader, Muhammad Yunus, said in an interview last weekend with The Sunday Times that Ms. Siddiq should apologize and that the London properties should be handed back to his government if they were gained through corrupt means.

On Tuesday, Kemi Badenoch, leader of the main opposition Conservative Party, criticized the prime minister for failing to take action earlier.

“It was clear at the weekend that the anti-corruption minister’s position was completely untenable,” she wrote on social media. “Yet Keir Starmer dithered and delayed to protect his close friend.”

In the official exchange of letters following Ms. Siddiq’s departure, Mr. Starmer wrote that his former minister had “made a difficult decision,” but added that “the door remains open” to her going forward — signaling that a return to a government might be possible for her in the future.

Mujib Mashal contributed reporting.