They stayed up late studying for civics tests. They went to classes, paid hefty fees and underwent background checks.

During the last year, nearly 1 million legal immigrants applied to become American citizens, many of them hoping to take the oath of citizenship in time to cast their first ballots on Nov. 8 in a presidential race where immigration has been fiercely debated.

But as the number of aspiring citizens grew this year, the backlog at the federal agency that approves naturalizations swelled. With the agency now reporting that it takes up to seven months to complete the process, Obama administration officials are reluctantly admitting that many — perhaps most — of the immigrants in the backlog will not become citizens in time to vote.

“I’ve been checking my mail every day, but I haven’t heard anything,” said Francisca Fiero, 73, a Mexican immigrant in Las Vegas. “I’m starting to get very worried.”

Fiero, who has had a green card for a decade, applied in January and gave her fingerprints in June. Since then, nothing. The voter registration deadline in Nevada is Oct. 18.

In the last year almost 940,000 legal immigrants applied to become citizens, a 23 percent surge over the previous year. As of June 30, more than 520,000 applications were waiting to be examined, a pileup that increased steadily since last year.

Immigration officials “anticipated that there would be a spike in applications this year, but the increase has exceeded expectations,” said Jeffrey T. Carter, a spokesman for the U.S. Citizenship and Immigration Services, the agency in charge of naturalizations.

The official figures revealing the backlog, published in late September, came as a shock to immigrant groups that put on a nationwide push early this year to help eligible immigrants to naturalize. Some of the biggest increases in applications came in battleground states where they had focused their efforts, including a 30 percent increase over a year earlier in Colorado, a 40 percent increase in Florida and a 53 percent increase in Nevada.

“The agency has developed an acute case of the slows, and it could not be a more critical moment,” said Tara Raghuveer, deputy director of the National Partnership for New Americans, a coalition of 37 groups that held citizenship workshops around the country. The groups scrambled to file applications before May 1, she said, after the immigration agency originally advised them that the process would take four to six months.

This year for the first time the naturalization drive also had high-profile backing from the White House, which sponsored ad campaigns, gave $10 million to community groups and made fixes to make it easier to apply. But officials said the White House was not monitoring the results to confirm that the immigration agency was completing naturalizations in a timely way.

In the presidential race between Hillary Clinton and Donald Trump, the immigrant vote could be pivotal, especially in states with large numbers of Latino immigrants.

The naturalization backlogs are bad news for Clinton because polls in those states show Latinos favoring her over Trump by gaping margins. In Florida, for example, more than 66,000 potential new voters stuck in the backlog could be enough to affect the outcome of a race that polls show is a virtual tie.

Citizenship applications generally surge during presidential election cycles. In 2007 and 2008, more than 1.4 million immigrants became citizens, stirred by a combination of an impending fee increase and the historic candidacy of Barack Obama.

This year, immigrant groups set a goal of 1 million naturalizations, and the application numbers were on track to reach it. In May, the Obama administration announced another fee increase for later this year, which could have moved some latecomers to apply. Then the backlogs emerged.

Many immigrants decided to become citizens because they just wanted to vote, without being drawn to a particular candidate.

“I want to put in my voice to be heard just like everybody else,” said Geraldine Rolle, a 65-year-old immigrant from the Bahamas, who has been a legal resident since 1991 and now lives in Fort Lauderdale, Fla. She worked for years as a housekeeping supervisor in Miami hotels.

She applied in February. After having her photograph taken in April, she heard nothing more. “They keep telling me you have to be a citizen to vote, and now it doesn’t look like I’m going to get that before the elections,” Rolle said sadly.

Fiero, in Las Vegas, has had a rocky time. She came to the United States as a legal resident and has always kept her green card up to date. A cafeteria worker in an elementary school, she applied to naturalize in January. But the agency rejected her application as incomplete.

When many similar rejections were reported in Nevada, the Culinary Workers Union, which helped Fiero and many others to apply, appealed to Sen. Harry Reid. Immigration officials acknowledged to Reid that the applications had been denied by mistake. Fiero was allowed to apply again a month later.

Like many Latino immigrants, Fiero said she was eager to vote because she does not like Trump and is a fan of Clinton’s. “She seems like the right person to help poor families get better houses,” Fiero said.

Officials at the citizenship agency said they shifted employees to offices where the workload had mounted and authorized overtime for many offices. But Carter, the spokesman, said that nationwide the agency was “still within our normal processing times.”

Internal emails from the immigration agency’s Houston office, which were recently leaked by Republican leaders in the Senate, show it was working to pick up the pace. A message on July 21 exhorted employees to finish as many naturalizations as possible before Oct. 1 “due to the election year needs.” The Houston office opened on two Saturdays that month to try to address the backlog.

But in New York, citizenship groups have heard from anxious immigrants who have waited as long as a year with no end in sight. In Nashville, city officials had to cancel a swearing-in ceremony planned to take place this weekend during a festival. The judge scheduled to conduct it reported no immigrants were ready. Immigration officials said other demands on agency employees’ time got in the way.

The agency is also facing pressure to slow down. In a letter last week, Sens. Charles Grassley of Iowa and Ron Johnson of Wisconsin, both Republicans, cited a recent case in which more than 800 immigrants who had been ordered deported were granted citizenship because of gaps in the agency’s vetting.

Rather than speeding up, the agency “should instead be putting on the brakes,” the senators wrote.

But at the Culinary Workers Union in Las Vegas, immigrants are fuming. “It really is outrageous that people can do everything right and still be denied an important right as a citizen,” Yvanna Cancela, a union official, said.

© 2016 The New York Times Company