“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 J. Trump, the immigrant vote could be pivotal, especially in states with large numbers of Latino immigrants.

The naturalization backlogs are bad news for Mrs. Clinton because polls in those states show Latinos favoring her 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 one 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.