For New York immigrant families just scraping by, the $680 it costs to apply for United States citizenship may be too much of an obstacle.

Enter a new kind of lottery, one with a potential lifetime payoff.

Gov. Andrew M. Cuomo, a Democrat, plans to announce on Wednesday a yearlong initiative that will cover the application fee for about 2,000 lottery winners who are not destitute enough to qualify for a waiver of the fee but are still considered to be working poor.

“From my experience in the Latino community, people love lotteries, people love contests, and so this allows us to bring attention to the issue and gets people excited,” Rossana Rosado, the New York secretary of state, said. “And while we’re there, we can tell them, ‘This is not as hard as you think it is, and we have help to get you through this.’”

An estimated 160,000 residents of New York State who hold a green card and whose household income falls between 150 percent and 300 percent of the federal poverty line will be eligible for the lottery, the state said.

The lottery will be divided into two phases. Five hundred vouchers will be distributed across the state in the next several months, with the remainder distributed in 2017. The application fee is expected to increase to $725 this fall.

While it may not have the draw of Powerball, the program, called NaturalizeNY, could help raise new citizens’ incomes and tax revenues for the state.

“Citizenship allows you the ability to change your lives, to qualify for state jobs, government jobs, things that take people squarely into the middle class and beyond,” Ms. Rosado said.

Private donors will match public money for the $1.25 million program, the first of its kind in any state, Ms. Rosado said.

Nonprofits that serve immigrants have seen how discouraged they are by the cost of becoming a citizen, and by the language requirement. As part of NaturalizeNY, the state will team up with organizations to offer free legal services, plus English classes and civics lessons to help green-card holders pass the citizenship exam.

“To have this application paid for will be a huge boost for people who are interested but just couldn’t afford to take part,” said Jo-Ann Yoo, executive director of the Asian American Federation, a pan-Asian group based in Manhattan. “It’s more important than ever, given the anti-immigrant narrative that we’re hearing.”

The state’s Office for New Americans, which was founded in 2013, has enlisted nonprofit groups to provide immigration services at 27 “opportunity centers” across the state.

Organizations from a number of communities are taking part. The Asian American Federation, the Manhattan-based Chinese Progressive Association and the Council of Peoples Organization in Midwood, Brooklyn, which offers language and citizenship classes for Muslims, Arabs and South Asians, are working together. They will sponsor joint citizenship clinics at the Council of Peoples’ offices in August.

On Long Island, one citizenship drive at two locations of the Central American Refugee Center brought in 115 applications; the program’s goal was 125 for the year.

Patrick Young, the center’s program director, said that with demand high because of the presidential election, the group sent out more than 1,000 naturalization applications this year, but that there were still an estimated 100,000 people on Long Island eligible to become citizens who had not yet applied.

Of the 27 opportunity centers, four are on Long Island, 15 are in New York City, and the rest are in smaller cities like Utica and White Plains.

While it may be too late to become a citizen in time to vote in this year’s presidential election, “there is always another election,” Mr. Young said.

Advocates and state officials hoped the program would continue in future years. With funding from the Robin Hood Foundation and the New York Community Trust, researchers from the State University of New York at Albany, Stanford University and George Mason University are studying how cost factors into naturalization.

Allan Wernick, the director of CUNY Citizenship Now, which provides legal services to immigrants at centers across the city and also runs an opportunity center, applauded the initiative. However, he said the need for it pointed to a larger issue. “This is a great program for the people who benefit from it,” Mr. Wernick said. “But a greater concern is that the fees are too high.”