Like many Israelis, Bronx-born Eli Knoller, who has dual citizenship and lives in an Israeli settlement in the West Bank, hopes the next American president allows Israel to “finish the job” against Hamas in Gaza, where his son was killed in battle.

Abduljabbar Alqam, a Palestinian American who lives just a few miles away, is horrified by what he calls U.S. complicity in the suffering of Palestinians in Gaza.

But they have at least one thing in common: Neither planned to vote for Vice President Kamala Harris. Mr. Alqam believes the Biden administration has been too supportive of Israel and the war in Gaza; Mr. Knoller believes it has not been supportive enough.

In the Israeli-occupied West Bank, tens of thousands of U.S. citizens live on opposing sides of one of the world’s most entrenched conflicts, and many are bitterly disappointed with the Biden administration’s approach to Israel’s war in Gaza, triggered by the deadly Hamas-led attack on Israel last October.

Opinion polls show that Israelis largely support former President Donald J. Trump, fondly recalling his near-unreserved support for a country now facing increasing international isolation, while many Palestinians are frustrated with President Biden’s backing for Israel and see little difference between the two candidates. Their frustrations reflect the wider discontent over the war in Gaza across the American political spectrum.

“The Democrats need to lose, and they need to know that one of the biggest reasons they lost is their stance on Israel,” said Mr. Alqam, 37, who planned to vote for the Green Party candidate, Jill Stein. “It’s about making a statement.”

The war in Gaza has confounded the final year of Mr. Biden’s presidency, creating a rift inside his party and exposing American weakness in the Middle East. His envoys have shuttled around the region for months trying in vain to clinch a cease-fire deal.

How his successor will affect the conflict is far from clear. Mr. Trump took staunch pro-Israel stances during his term, including a proposed peace plan that strongly favored Israeli demands over Palestinian ones. But he has also called on Israel to wind down the war.

Ms. Harris has mostly stuck to President Biden’s views: backing Israel’s right to self-defense while pressing for a deal to end the war and release the hostages held in Gaza. She has taken a stronger tone on Palestinian suffering, but has not signaled a markedly different approach if elected.

Estimates vary widely, but at least 150,000 Americans live in Israel, which has a population of roughly 10 million, according to the U.S. government. Roughly 60,000 live in Israeli settlements in the West Bank — which much of the international community deems illegal — making them roughly 15 percent of the settler population, according to Sara Yael Hirschhorn, a lecturer at Haifa University.

Thousands of Palestinian Americans also live in the West Bank, though there are no official statistics.

Generations of Palestinians have ferried back and forth between the United States, where they are equal citizens under the law, and the West Bank, where they are subject to Israel’s two-tiered system. Israeli law gives settlers all the rights of their neighbors in Israel proper, while Palestinians in the West Bank — U.S. citizens or not — live under Israeli military occupation, with far fewer rights.

Kory Bardash, the co-chairman of Republicans Overseas Israel, lives in Efrat, a settlement with many Americans, where he coached Little League baseball. Over the past year, the United States has projected impotence rather than decisiveness in the Middle East, which is bad for Israel, he said.

“Under the current administration, the players in this neighborhood sense weakness,” said Mr. Bardash, who canvassed for Mr. Trump.

A handful of settlers, pointing to Mr. Trump’s often unpredictable political zigzags, still support Ms. Harris, including Herzl Hefter, an American-born Orthodox rabbi who lives in Efrat. He said at least some of his neighbors shared his misgivings over what he called Mr. Trump’s “moral rot” but had nonetheless decided to “hold their nose and vote for Trump.”

“But it doesn’t mean that in this policy or that policy, maybe Trump would be better,” said Mr. Hefter, 67. “It’s really impossible to know, because he’s totally unreliable and crazy.”

A few miles north of Jerusalem, thousands of Palestinian Americans live in towns where many split their time between the Middle East and the United States. In Turmus Aya, a quiet, relatively prosperous village close to Ramallah, Americans make up a large part of the population, particularly in the summer, when expatriates pack into the town, towing their children for monthslong visits.

Mr. Alqam, who was born in New Jersey, spent several childhood years living in Turmus Aya. In 2023, he and his wife moved back from Louisiana so his three children would connect with their roots and learn Arabic — although many children in Turmus Aya prefer to chatter together in English.

Returning from abroad brought Mr. Alqam again face-to-face with the maze of Israeli restrictions on Palestinians across the West Bank, which Israel says are necessary to prevent further militant attacks. He sought to reassure his children that their U.S. passports might protect them regardless.

“In America, we would have equal rights. But in this country, they have superiority, more rights, more protection, more safety,” said Mr. Alqam, referring to Jewish Israelis.

Two weeks after their arrival, Jewish extremists stormed into their hometown, torching homes in retaliation for a Palestinian attack earlier that day that had killed four Israelis. One of the town’s residents was fatally shot during the clashes.

Mr. Alqam conceded that Israeli hard-liners might be further emboldened if Mr. Trump was elected, potentially moving to annex the West Bank. But he said the situation was rapidly getting worse either way — making it important to first change attitudes in the United States.

“I’m willing to take four years of a little bit more suffering in order to hopefully change something bigger,” Mr. Alqam said, referring to a potential Trump victory.

Some Palestinians do support Ms. Harris, while acknowledging frustration with her on Gaza. Hakeem Asheh, a Palestinian American living in the West Bank city of Nablus, said he was willing to “give Harris a chance.”

“The Democratic Party is changing its views on the Israeli-Palestinian conflict. There’s a new vision,” said Mr. Asheh, who worked as a drugstore manager in Connecticut before returning to Nablus a decade ago. “I doubt Harris will implement it — but over time, that might change.”

But it is unclear how much either candidate would influence Israel, where many view the ongoing wars as existential conflicts.

Mr. Knoller, 60, moved to Karnei Shomron, a settlement in the northern West Bank which, like Efrat, was built up in part by American immigrants. In July, his son Nadav, 30, was killed while on a third tour of reserve duty in Gaza, leaving behind a wife and 18-month-old son.

For Mr. Knoller, the decision to vote for Mr. Trump was simple.

If Ms. Harris is elected, the United States will probably “pressure Israel to possibly reach a cease-fire and release thousands of Hamas terrorists” in exchange for Israeli hostages held in Gaza, he said. “That’s something I can’t support.”

The Kremlin may quietly prefer that Donald J. Trump beat Vice President Kamala Harris in the U.S. presidential election on Tuesday. But among hard-liners and ordinary Russians, hopes that either candidate could help bring a swift end to the war in Ukraine — by stopping military aid and effectively forcing Kyiv to accept Russia’s occupation of its territory — are low.

And whatever the results, the prospect of improved U.S.-Russia relations seems even more distant.

“We don’t have anyone to root for,” Dmitri Kiselyov, an anchor at the state-run Channel 1 TV station, said on a Sunday show. “That’s why we are just calmly observing,” he asserted, even as the U.S. authorities recently accused Russian operatives of using disinformation to try to sway the election.

In 2016, pro-government officials in Russia’s Parliament celebrated Mr. Trump’s victory by popping dozens of bottles of Champagne.

At that time, there was a genuine belief that the Republican political newcomer could overturn U.S. sanctions against Russia for its illegal annexation in 2014 of Ukraine’s Crimean peninsula, make changes to the global order by paying less attention to democratic principles and human rights ideals, and, perhaps, even lavish President Vladimir V. Putin with the type of respect he believed he deserved as the leader of a country with 11 time zones.

Eight years later, members of the Russian ruling class are more pessimistic.

“The elections will not change anything for Russia since the candidates’ positions fully reflect the bipartisan consensus on the need for our country to be defeated,” Dmitri A. Medvedev, a former president and prime minister of Russia who now serves as deputy chairman of the Security Council of Russia, wrote on Monday on Telegram.

That sentiment is prevalent in a Russia that feels scorned and underestimated by Democrat administrations and betrayed by a President Trump who openly praised Mr. Putin but in 2018 reversed Obama-era prohibitions on selling anti-tank weapons to Ukraine.

“Officials in the Kremlin were screaming at one another in shock,” remembered Aleksei A. Venediktov, the former editor of Ekho Moskvy, a popular radio station that the government shut down after the invasion of Ukraine. “The sense of betrayal was palpable.”

The U.S. vote comes at a critical juncture for Ukraine as it struggles to recruit enough troops and maintain financial and military support from Western allies.

Russia has gained an upper hand on the battlefield, American military and intelligence officials have concluded. But Moscow may also face its own troop shortages as the end of a third year of fighting nears. Should he win, Mr. Trump has promised to end the war before his inauguration, without describing how or saying that he wants Ukraine, America’s ally, to win.

“They respect me,” he has said of Mr. Putin and his Ukrainian counterpart, President Volodymyr Zelensky. His running mate, JD Vance, has put forward a proposal that would have Russia hold on to most of the Ukrainian territory it occupies, keep Ukraine militarily neutral and demilitarize the current front line.

“Trump is someone who is both pragmatic and lacking principles,” said Konstantin Remchukov, a Moscow newspaper editor with Kremlin connections. “When a person has no principles and is pragmatic, it’s easy to make a deal with them. Anyone without principles, but a pragmatist, is a much better and more comfortable negotiating partner than a person with principles, especially principles declared in public,” he said, alluding to Ms. Harris and her Democratic Party’s pro-democracy, human rights agenda.

“Trump, even for those who think he’s nuts, they still understand him better than Kamala, who for many Russians is kind of a hologram of what America wants to think of itself,” said Nina Khrushcheva, a professor of international affairs at the New School in New York, who researches propaganda and also lives in Moscow.

“They understand his narratives; they understand his conservatism,” she added. “But I don’t think anybody believes that he can end the war in Ukraine.”

Even pro-government Russians are aware as they watch the U.S. vote that they have not had a genuine choice or nail-biter of a presidential election at home in over two decades. Many in Moscow say they would gladly support any U.S. candidate who helps stop the war in Ukraine.

But Alyona, a 27-year-old Muscovite, said, “We all agree that none of them seem able to do it — definitely not Trump.”

Alyona, who insisted her last name be withheld because of a fear of speaking to a Western newspaper, had little hope that Mr. Trump could make a difference in relieving Russians of the heavy sanctions imposed since 2022.

Mr. Medvedev, who has derided Ms. Harris as “stupid, inexperienced and controllable,” said he did not have faith in the Republican candidate to end the war, either. Mr. Trump would “be forced to comply with the system’s rules” and “will not be able to stop the war — not in a day, not in three days, not in three months.”

The only thing that mattered is how much money Congress is willing to spend on supporting Ukraine’s military, he said, using a vulgar expression.

“That’s why the candidate the Kremlin really wants is Mr. or Mrs. Chaos,” said Mr. Venediktov, the former radio station editor. “Their ideal candidate is whoever brings chaos, whatever makes the U.S.A. weaker in terms of domestic politics. If the result brings people to the streets, mires the political process in court proceedings, anything that pushes Ukraine down the U.S. agenda, it can be considered good for Russia.”

Some observers say Russian leaders are playing down any support for Mr. Trump not only because of a sense of betrayal, but also not to seem eager for a Republican victory, considering the Kremlin’s well-documented history of interference in 2016 to shift the results in Mr. Trump’s favor and new allegations of Russian efforts to influence the vote on Tuesday and cast doubt on its result.

Mr. Putin made seemingly sarcastic remarks in support of Ms. Harris in September, praising her “infectious laugh” and saying that since President Biden was out of the race, “we will support her.”

More recently, he said he found Mr. Trump’s “desire to do everything to end the conflict in Ukraine” to be “sincere,” without commenting on Ms. Harris’s Ukraine policy.

But as he made those comments during a news conference last month at the BRICS summit, he made clear that relations between Washington and Moscow, which are at their lowest point since the end of the Cold War, could be restored only by giving Russia the respect on the global stage that Mr. Putin believes it deserves.

“How Russian-American relations will develop after the election will depend on the United States,” he said. “If they are open, then we will also be open. And if they don’t want it, then fine.”

Mr. Kiselyov, of Channel 1’s flagship weekly program, “News of the Week,” has used the U.S. campaign as proof of America’s deep polarization, moral decline and potential for political violence.

“Whoever is declared the winner, the other side will definitely not accept this, accusing the opponent of falsification,” he said on the Sunday night program, declaring: “This is a harbinger of the riots that will ensue.”

Citing a CNBC article about high-net-worth Americans, he added: “A record number of wealthy Americans expressed a willingness to leave the country indefinitely right after voting day. The reason? Fear.”

There is anxiety and fear among Americans across the political spectrum about the vote, a mood that the Kremlin hopes to capitalize on not only for its war aims, but also to telegraph to millions of Russians that as chaotic as two and a half years of war and sanctions may be, things are out of control in the United States, too.

Nataliya Vasilyeva contributed reporting.

As she surveyed her home in Hong Kong, Liu Lanhua tried not to be bothered that her narrow kitchen doubled as the family’s only bathroom.

Colanders, pans and hairbrushes dangled above the toilet. Jars of chili oil were precariously balanced on water pipes. A stew of chicken wings and chestnuts warmed on an electric stove a few feet from the shower faucet.

She and her 12-year-old daughter are among 220,000 people in Hong Kong living in subdivided homes, which have long been among the starkest examples of the city’s vast income inequality.

Now her home is under threat. Hong Kong’s leader, John Lee, last month announced that the city would impose minimum standards on the size and fixtures of such apartments. The policy is expected to phase out more than 30,000 of the smallest subdivided homes.

In Ms. Liu’s home, there was no space for a sink; the only spot for two pet turtles was in a basin under the fridge. “If we had money, these would be in separate rooms,” she said, looking at the cluttered kitchen and toilet.

Beijing has urged the Hong Kong government to get rid of subdivided units and other tiny homes by 2049, because it regards the city’s housing shortage as one cause of the antigovernment unrest of 2019.

But Mr. Lee’s plan has raised concerns among experts and advocates of more public housing, who say it would raise already high rents for the poor and evict a number of people without clear plans for their resettlement. It also doesn’t address the worst types of housing in the city: rental bed spaces so small they are known as coffin, or cage, homes.

Of Slums and Slumlords

Hong Kong’s subdivided homes, created when apartments are carved into two or more units, are usually in old tenement buildings in densely packed, working-class neighborhoods. Despite their often dilapidated conditions, the units are in high demand because affordable housing is in short supply.

Hong Kong has among the world’s most expensive homes, and highest rents. The average living space per person is 64.6 square feet — less than half the size of a New York City parking space. Owners of tenement apartments partition the units into smaller ones to rent them to more people.

“These are effectively slums and the landlords are slumlords,” said Brian Wong, a researcher at the Liber Research Community, an independent group in Hong Kong focused on land use and urban issues.

He added that the landlords who rent out subdivided units are often upper-middle-class residents looking to maximize profits. Paradoxically, the rent price of such units, on a per-square-foot basis, is usually higher than that of larger private apartments.

Ms. Liu pays $500 a month for her home of about 80 square feet, about a quarter of what she earns working at a construction site. Her unit is in a 60-year-old tenement building with peeling pink and yellow paint in Kwun Tong, a district in east Kowloon that was once an industrial heartland, with cotton mills and a soy sauce factory.

“I will live where it’s cheap,” she said, adding that she wanted to pay for after-school classes for her daughter. She has been waiting for six years to move into public housing but has no idea when that might happen.

Ms. Liu and her daughter sleep on bunk beds in the 60-square-foot main room, pushed against windows that are covered with paper for privacy and always closed to keep rats out. Ms. Liu appreciates that her neighbors don’t complain when her belongings spill into common spaces.

Kwun Tong is the most densely populated district in Hong Kong, and the poorest. People are drawn to it for its connectivity and services. Ms. Liu moved there six years ago to take a housekeeping course. Her daughter rides two stops on the subway to attend public school and studies with a tutor nearby until dinnertime. Their apartment is close to a large wet market.

The Hong Kong Leader’s Plan

Ms. Liu’s home would not meet the standards required under the policy outlined by Mr. Lee, the city’s chief executive, which stipulates that each home must have a separate bathroom and kitchen. It would likely require significant renovation or remodeling.

The policy also calls for apartments to be at least 86 square feet and come with windows.

Ms. Liu’s bathroom and stove are in a narrow cubicle that is slightly more than 20 square feet, separated from the main room by a common hallway. There is one faucet but no shower cubicle or sink, so she soaks ingredients in a bowl on the floor. The fridge faces the toilet.

Merged toilet and kitchen setups like this are common in subdivided apartments. Some apartments come only with toilets or kitchens that are shared with other households.

The government estimates that 30 percent of the city’s 110,000 subdivided homes will fall short of the new standards.

The Housing Bureau said in a response to questions from The New York Times that the rules were needed to improve living conditions. It said it would inspect apartments and that landlords could face prison time for not complying with the rules.

The bureau also said that landlords would have a few years to renovate their units to meet the standards, and register them in a centralized system.

Plan Leaves Much to be Desired

At a recent meeting between social workers with the Kwun Tong Subdivided Home Concern Group, a nonprofit, and residents of the district, questions were raised about the government’s plan. What are the standards for a proper toilet? If rents go up, will the government provide tenants with subsidies? Will those evicted be given priority in housing wait-lists?

“The standards have been raised but our finances haven’t,” said Moon Tang, a mother of three. She also wondered what would happen to people if they were evicted. “If they had money, they would have rented a more expensive space in the first place,” she said. “Where do they go?”

In its emailed response to questions, the Housing Bureau said the government would “adopt a gradual and orderly approach” to the changes and would help residents “where necessary.” Most affected tenants would be able to turn to an increased supply of permanent and temporary public housing apartments by the time the rules come into force in the coming years, it said.

Experts note, however, that the new policy also fails to address problems faced by those living in “cage homes” or “coffin homes” — bed spaces separated by wired metal or panels of wood. (Such spaces are regulated by a separate law.)

Siu Ming Chan, an assistant professor at the City University of Hong Kong who researches poverty and housing, said the rules could lead to a rise in rents, making apartments even more unaffordable. The government should increase subsidies for those affected by the policy, many of whom are older and live alone, he added.

Ben Shek, 68, a former technician who lives alone in a 60-square-foot Kwun Tong apartment that would likely be considered substandard, does not want to move. He suffered a stroke more than a decade ago that left him with a limp and unable to work. He shares a bathroom with two other families, inside a carpentry workshop. He likes his place because it is on the ground floor, making it easy for him to get around.

“Since I’m not working anymore, I don’t get to have too many expectations,” he said. “And even if I did, they can’t be too high.”

Britain’s Conservative Party made history last weekend, becoming the country’s first major party to elect a Black woman as leader.

And yet this milestone was reached not by a party of the progressive center-left but by Britain’s oldest and most traditional conservative political force.

“I’m glad because it shows that my country and my party are actually places where it doesn’t matter who you are or what you look like,” the new leader, Kemi Badenoch, told the BBC on Sunday. After all, the Conservatives have broken glass ceilings before, picking all three of Britain’s female prime ministers, as well as its first leader who wasn’t white, Rishi Sunak.

Now 44, Ms. Badenoch was born in London and raised in Nigeria, returning to Britain when she was 16. In her first parliamentary speech, in 2017, she declared, “To all intents and purposes, I am a first-generation immigrant.”

She will now lead a party that has struck an increasingly harsh tone on immigration, a shift that mirrors her own changed views on the subject.

Her rise to the top echelon of British politics is the result partly of determination, hard work and a fearlessness that have helped her survive the at times brutal infighting within her party in recent years. Claire Coutinho, a fellow Conservative lawmaker, wrote on social media that “Kemi’s fierce intellect and love of Britain will make her an excellent leader.”

But she has also benefited from good fortune and a concerted effort by the former Prime Minister David Cameron to diversify the Conservative Party almost two decades ago.

“The reason she has been able to advance so rapidly through the party is the politics that David Cameron put in place — because he felt that the Conservative Party didn’t look enough like modern Britain,” said Robert Ford, a professor of political science at the University of Manchester.

The irony is that Mr. Cameron’s brand of liberal Conservatism has ended up benefiting Ms. Badenoch, a staunch opponent of so-called “woke” diversity policies. Mr. Cameron’s deliberate recruitment of parliamentary candidates who were female and not white “is the kind of thing she is scathingly critical of,” Professor Ford added.

Plain-speaking and sometimes antagonistic, Ms. Badenoch has a reputation for wading into debates on identity politics. Her pitch to become party leader was that she is “Labour’s worst nightmare because they can’t portray me as prejudiced,” an apparent reference to her racial identity.

Less well scripted were her comments, during the leadership campaign, suggesting that maternity pay was excessive and joking that a minority of civil servants performed so poorly that they should be in jail. Both were seized on by critics as the kind of unforced errors that could prove damaging to the Conservatives if they want to win back centrist voters.

Ms. Badenoch is a “high risk, high return” leader, said Professor Ford. “She is prone to say things that can blow up in her face and is not so quick to row back on them” he said. But on the plus side, he said: “She gets attention.”

Ms. Badenoch was born in 1980 to a mother who was a lecturer in physiology and a father who was a doctor. Her parents were prosperous enough for their eldest daughter, Kemi, to be born in a private hospital in Wimbledon, in southwest London, before returning to Lagos, the Nigerian city where she was raised.

Initially she lived a comfortable life but, after political and economic upheavals, her family’s fortunes declined. Years later, she recalled “doing my homework by candlelight because the state electricity board could not provide power,” and fetching water from a nearby well when the faucets ran dry, because of failures by the nationalized water company. These experiences contributed to her preference for a small state. “I was unlucky enough to live under socialist policies. It’s not something I’d wish on anyone,” she wrote.

At 16, Ms. Badenoch returned to London to stay with friends, working part time at McDonald’s as she studied. She has recalled being disappointed by her teachers’ low expectations but went to Sussex University to study computer systems engineering.

Asked later what formed her Conservative views, Ms. Badenoch pointed to her experience at college “among snotty, middle-class north Londoners” who talked of helping Africans. “These stupid lefty white kids didn’t know what they were talking about,” she said in an interview this year with The Times of London, adding that she believed a campaign to boycott Nestlé over its promotion of infant formula in African countries was ill informed. “That instinctively made me think, ‘These are not my people.’”

Sunder Katwala, the director of British Future, a research institute, described Ms. Badenoch’s rise as “a remarkable personal achievement,” adding that she was “incredibly formed by the experience of being a young Black African woman who comes to Britain, and a lot of that strengthens her to become conservative.” Ms. Badenoch, he said, is very positive about British democracy because she is comparing it to the Nigeria she grew up in — “whether the lights are on or not, or whether you have to pay bribes.”

When Mr. Cameron became leader of the Conservatives in 2005, he concluded that the Tories were “the oldest political party in the world — and we looked it.” Intent on diversifying the party, he oversaw a policy in which party members were encouraged to pick parliamentary candidates from a curated list of candidates, half of whom were female and a large proportion of whom were Black or from ethnic minority backgrounds.

In 2010, Ms. Badenoch, then Kemi Adegoke, ran for election in a pro-Labour district in south London where she unsurprisingly lost. But the campaign brought her close to the man she was to marry, Hamish Badenoch, a privately educated Conservative activist, now a managing director at Deutsche Bank.

In interview, Ms. Badenoch has acknowledged that her reputation for being confrontational extends to home life. “He thinks my capacity to tolerate conflict is too high,” she said of her husband. “He says, ‘You’re the politician in the family and I’m the diplomat.’”

A supporter of Brexit, Ms. Badenoch prospered under three Conservative prime ministers and was promoted to the cabinet in 2022.

Her position on immigration has hardened in recent years. In comments from 2018 that recently resurfaced, she welcomed the Conservative government’s proposal to relax restrictions on visas for skilled migrants. But net migration has trebled since Brexit, and she said she had since changed her mind.

Ms. Badenoch now argues that “numbers matter but culture matters more,” adding that the most important fact was who was coming to Britain and what their values and aspirations were.

She has also been critical of diversity policies. “Many practices have not only been proven to be ineffective, they have also been counterproductive,” she wrote earlier this year.

“She sees critical race theory as very dangerous, and she thinks that Britain’s history on race is very different to America’s and that we don’t want America’s race politics,” Mr. Katwala said.

Part of Ms. Badenoch’s political appeal is her avoidance of the bland platitudes favored by many politicians. But at times her trenchant views have upset some colleagues, and many analysts had assumed that she would fail to make the final shortlist for the Conservative leadership.

Under the rules of the contest, the party’s lawmakers choose the final two candidates before dues-paying party members — around 130,000 people — vote for the winner.

When the lawmakers voted on the final two in October, the favorite, James Cleverly, a more centrist former foreign secretary, was unexpectedly eliminated. That left Ms. Badenoch competing against Robert Jenrick, another right winger.

Analysts think that some of Mr. Cleverly’s supporters lent their votes to Mr. Jenrick to push him into the final runoff instead of Ms. Badenoch, but the tactic backfired.

“One of the reasons she’s leader is that so many of the members of Parliament she now leads were so eager to keep her off the ballot that they ended up knocking off the one they wanted to keep on the ballot,” said Professor Ford. “They tried to be too clever by half.”

The result of that miscalculation is that British politics is entering a new, unpredictable era. While Ms. Badenoch has shown herself prone to occasional gaffes and unpopular opinions that could limit her appeal to the broader electorate, her confidence and energy may create a challenge for Prime Minister Keir Starmer.

Some senior Labour lawmakers have also noted that her leadership casts an unflattering light on the relative lack of diversity in their own party’s leadership team. Labour has typically presented itself as more progressive on race issues.

“I think her identity and heritage makes for a very visible symbolic contrast at prime minister’s question time in Parliament: an older white man facing a younger Black woman,” Professor Ford said. “It’s interesting, it’s novel — it will pose new questions for Labour.”

The Biden administration’s support for Israel in the war in Gaza has been divisive for left-leaning voters in the United States, including many Arab Americans, and some say it has soured them on Vice President Kamala Harris’s candidacy.

Many in Gaza share that anger over the United States’ willingness to keep shipping weapons to Israel to carry out its campaign against Hamas despite the death and devastation in Gaza. But in interviews across the territory, many said they were skeptical that either Ms. Harris or former President Donald J. Trump would do much to improve their situation.

“I am fearful that both candidates are for the same thing, which is no end in sight for the war in Gaza,” said Abdul Kareem al-Kahlout, 35, a math teacher in Deir al Balah.

The war began after the militant group Hamas led the Oct. 7 terror attack that Israeli authorities say killed about 1,200 people in Israel. Since then, the Israeli military’s bombardment and ground operations in Gaza have killed more than 43,000 people, according to local authorities, a figure that includes Hamas fighters. The war has pushed the remaining population to the brink of famine and left much of the territory in ruins.

Many people interviewed in Gaza said they were more focused on keeping themselves and their loved ones alive after more than a year of war. They have had little access to electricity or the internet, or to adequate food and medicine, so they have not had much time to follow American politics.

“I have no preference,” said Mohammed Owaida, 33, who is from Khan Younis in the southern Gaza Strip. “We only need one thing: for this war to come to an end. We are exhausted. Whoever wins and can do that, I support.”

Across the border, polls show that Israelis overwhelmingly view Mr. Trump as the candidate who best serves their country’s interests, an opinion based largely on the sense that his first term in office brought benefits to Israel. While a Harris win would offer a sense of continuity at a turbulent time, many Israelis assume it would come with more criticism of Israeli policies toward Palestinians.

Mr. Trump took staunch pro-Israel stances during his term, including a proposed peace plan that strongly favored Israeli demands over Palestinian ones. But he has also called on Israel to wind down the war. Ms. Harris has mostly stuck to President Biden’s views: backing Israel’s right to self-defense while pressing for a deal to end the war and release the hostages held in Gaza.

Israelis generally believe that whoever wins, there won’t be a serious change in relations with the United States, their most important ally. And many in Gaza agreed, saying it was unlikely that the United States would waver in its support for Israel.

Lina Rabah, 36, said she thought American leaders viewed the people of the Gaza Strip as little more than “a chess piece on their board.”

“All I want is for the United States to see us as humans, not just as numbers in a long conflict,” said Ms. Rabah, who has three children.

“If either Trump or Harris truly values human life and human rights, then they must use their power — not remarks or speeches to the media — to press for an immediate cease-fire,” she said.

Rima Swaisi, a journalist from Gaza City who works for Wafa, the official news agency of the Palestinian Authority, said she thought Ms. Harris was more likely to pursue an end to the war than Mr. Trump, who has been a strong supporter of Prime Minister Benjamin Netanyahu of Israel. She called Ms. Harris the “less worse” option, and said she would never forget that Mr. Trump “gave Jerusalem to Israel” by moving the U.S. Embassy to that city from Tel Aviv in 2017, in a break with decades of American policy.

“If we have to choose between the two devils, then anyone but Trump,” she said. Of Ms. Harris, she said: “I just hope she wins and most importantly does something differently toward the Palestinian people.”

But some in Gaza said the election did not present Palestinians with a less bad option.

Hanin Ashour, 33, said she had lost four family members since the war began, including two young children: Mariam, 8 months, and Omar, 2. American officials have often talked about human rights, she said, but now she blames them for the deaths of her loved ones.

She has become so disgusted by U.S. policies, she said, that she will not even use humanitarian aid from American organizations. To her, the idea of pinning her hopes on an American politician is absurd.

“I cannot even eat anything that comes from the country that killed my innocent family members,” she said. “So what — am I supposed to wait to hear from U.S. presidents who support Israel with missiles? How am I supposed to listen to them?”

Isabel Kershner contributed reporting from Jerusalem.

The Israeli authorities are investigating a civilian who has been working over the past year in the office of Prime Minister Benjamin Netanyahu and is suspected of illegally obtaining and leaking classified documents to the news media.

The documents helped support Mr. Netanyahu’s reasoning for adding tough new conditions for a cease-fire deal with Hamas over the summer, amid intense public pressure for a deal to release Israeli hostages and end the fighting in Gaza.

The case has roiled Israel, where critics have accused Mr. Netanyahu of torpedoing a deal to return hostages and of prolonging the war in Gaza for political reasons. Key members of his governing coalition had threatened to quit if he made concessions to Hamas.

On Sunday, an Israeli court partially lifted a gag order to identify Eliezer Feldstein, who was hired last year to work as a spokesman in Mr. Netanyahu’s office, as a suspect in the case. Three other suspects in the case are members of the military and security establishment, according to the court, and have not been publicly named.

The investigation has revolved around the publication and manipulation of real and purported intelligence information in media outlets abroad, according to Israeli news reports and to an Israeli official who was not authorized to discuss sensitive information, including the case. The London-based Jewish Chronicle published — and then retracted — a report claiming Hamas was planning to smuggle Israeli hostages from Gaza to Egypt. A classified document that was leaked to the German newspaper Bild claimed that Hamas was trying to manipulate the Israeli public and wanted to draw out the negotiations.

How did this all get started?

On Sept. 1, the Israeli military announced that six Israeli hostages had been found dead in a tunnel in Gaza after being fatally shot by their captors, prompting a surge of mass protests and a wave of national anger and grief.

About 100 people taken captive by Hamas on Oct. 7, 2023, remain in Gaza. At least a third of them have been declared dead by the Israeli authorities.

On Sept. 2, in a televised news conference, Mr. Netanyahu presented his arguments for a new condition for a cease-fire deal with Hamas: Israel must maintain a permanent presence in the Philadelphi Corridor, a strip of land along Gaza’s border with Egypt. Without that presence, Mr. Netanyahu said, Hamas could smuggle hostages across the border into Egypt’s Sinai Desert, and from there to Iran or Yemen, where he said they could disappear forever.

He also displayed a handwritten document in Arabic that he said was the work of high-ranking Hamas members, which he said had been found in January by Israeli soldiers in an underground command post in Gaza.

The document contained instructions for increasing the psychological pressure on Israel by issuing videos and images of hostages and casting doubt on the Israeli government’s narrative that its ground operation in Gaza would help release the hostages. Mr. Netanyahu said it showed Hamas’s strategy of sowing internal discord in Israel and suggested that the popular protests against his government played into Hamas’s hands.

When did the leaks happen?

On Sept. 5, soon after Mr. Netanyahu’s news conference, The Jewish Chronicle, a British community newspaper, published a report by a freelance journalist. The journalist, Elon Perry, claimed he had obtained Israeli intelligence showing that the Hamas leader, Yahya Sinwar, was preparing to flee Gaza, via the Philadelphi Corridor, to Iran, and to take Israeli hostages with him.

The report cited intelligence gleaned from a senior Hamas official who was interrogated by Israel and from documents seized on the day the bodies of the six hostages were recovered.

Asked about the report in The Jewish Chronicle, Rear Adm. Daniel Hagari, the Israeli military’s chief spokesman, said that he was unaware of any such intelligence or plan by Mr. Sinwar.

The Jewish Chronicle later removed that story and others written by Mr. Perry from its website and ended its association with him. The incident cast a cloud on The Chronicle, a 180-year-old newspaper whose ownership has been shrouded in mystery.

On Sept. 6, a day after The Jewish Chronicle article ran, Bild published an article it said was based on a Hamas document laying out its plan for psychological warfare against Israel on the hostage issue, claiming that Hamas was in no rush to reach a deal or end the war. Some of the messaging was similar to points Mr. Netanyahu made in his news conference.

The Israeli military issued a statement on Monday saying it appeared that the document cited in the Bild article was found about five months ago and was “written as a recommendation by middle ranks in Hamas, and not by Sinwar,” as the Bild headline may have suggested. The document contained information similar to that found in earlier documents, the military said, adding, “The leaking of the document constitutes a serious violation.”

Critics say the exposure of the purported intelligence appeared to be part of a disinformation campaign by Mr. Netanyahu or by his supporters, intended to dampen the campaign for the hostages’ release and influence Israeli public opinion in favor of the prime minister’s negotiating positions.

Was Netanyahu involved?

Mr. Netanyahu has not been questioned about the allegations and his office has denied leaking information. Many details of the case remain murky because of the gag order.

In one of its first statements about the affair, Mr. Netanyahu’s office said on Friday that nobody from his office had been questioned or detained. On Saturday, the prime minister’s office offered a different version, saying that the suspect in question — later revealed to be Mr. Feldstein — had never participated in security discussions and had not seen or received classified information.

Mr. Netanyahu’s office also accused the authorities of carrying out a selective investigation, arguing that numerous reports based on leaked information had been published during the war without any consequences. It described the investigation as “aggressive and biased.”

How has the public reacted?

Israel has been in uproar over the revelations that seeped out during the weekend. Prof. Hagai Levine, an Israeli public health expert who is active in the campaign to bring the hostages home, wrote in a social media post on Sunday that “the hostages scam of Netanyahu’s office appears to be more serious than the Watergate affair that led to the resignation of President Nixon.” He described the allegations as a “combination of the abandonment of the abductees, breach of trust and the undermining of state security.”

What are the latest developments?

In the first official acknowledgment of the suspected security breach, a magistrate’s court in central Israel on Friday partially lifted the gag order on the case.

The court ruling stated that several people had been detained as part of a joint investigation by the Shin Bet domestic intelligence agency, the Israeli police and the military.

The judge, Menahem Mizrahi, said they were suspected of “a security breach due to the illegal transfer of classified information,” as well as putting sensitive information and sources at risk, and harming the chances of achieving the war’s objectives in Gaza.

In a subsequent ruling on Sunday allowing the publication of Mr. Feldstein’s name, the judge specified that the war objective he was referring to was the return of the hostages.

Ronen Bergman, Myra Noveck, and Johnatan Reiss contributed reporting.

The Israeli military raided Palestinian villages in the northern part of the West Bank early Tuesday, setting off clashes with militants. Four Palestinians were killed, according to Palestinian health authorities.

It was not clear whether the dead included militants or civilians and Palestinian authorities do not differentiate in their death tolls. But the Israeli military said it had engaged in firefights during the raids that killed militants in the village of Tamoun and that its aircraft had carried out strikes there and near the city of Jenin.

The armed wing of Islamic Jihad, an Iranian-backed militant group, said fighters in villages south of Jenin were firing bullets at Israeli forces and detonating explosive devices.

The raids in the Israeli-occupied West Bank suggested that Israel’s military was continuing to target armed fighters in the northern West Bank even as it conducts major operations in Gaza and Lebanon and braces for the possibility of a wider conflict with Iran.

Israel has been ramping up a crackdown in the occupied territory that began before the Hamas-led Oct. 7 attacks, with authorities increasingly concerned about bolder and more sophisticated attacks by Palestinians. The raids have left a swath of destruction in the territory, churning up roads and leaving many civilians scared to leave their homes in what Israel’s military says is a search for explosive devices.

Sadeq Nazzal, 60, an owner of a nursery in Qabatiya, not far from Jenin, said he heard a powerful explosion on Tuesday morning and described a chaotic scene, with military vehicles moving along the main north-south highway and sounds of gunfire in the distance.

“We’ve become used to this situation,” he said. “But every time it happens, it upends our lives. Workplaces and schools shut down.”

During a funeral procession held in Tamoun, one of the Palestinians killed on Tuesday had been wrapped in an Islamic Jihad flag. Palestinian militant groups often drape their fallen members in flags bearing their emblems, but they will occasionally claim unaffiliated people as being among their ranks.