New York officials are seeking additional rental assistance funding from the federal government after landlords and tenants criticized the state for slowly distributing the first round of funding this summer.

The state changed how it considers applications in the past six weeks, which led to the quick processing of 63,000 payments totaling $804 million, according to the state’s Office of Temporary and Disability Assistance. Another $1 billion has been promised to roughly 80,000 renters but not yet paid, the office said.

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New York officials are seeking additional rental assistance funding from the federal government after landlords and tenants criticized the state for slowly distributing the first round of funding this summer.

The state changed how it considers applications in the past six weeks, which led to the quick processing of 63,000 payments totaling $804 million, according to the state’s Office of Temporary and Disability Assistance. Another $1 billion has been promised to roughly 80,000 renters but not yet paid, the office said.

The federal government awarded New York around $2.6 billion to help tenants who have fallen behind on their rent during the Covid-19 pandemic avoid eviction. Lawmakers said the relief was necessary to prevent homelessness caused by lost income as a result of the pandemic.

Under New York law, both the tenant and landlord must participate in the application process, but discrepancies—including the legal name of a landlord—have snarled the process. A spokesman for the office said it is working with landlords to ensure that applications are completed, and has assigned more than 100 people to work on the problem.

The Treasury Department said this week it would reallocate funding from states and municipalities that hadn’t spent or obligated 65% of their original funding awards by Sept. 30. Brian Kavanagh, a Democrat from Manhattan who chairs the New York state Senate housing committee, said New York had cleared that threshold and is in a strong position to gain additional funding.

“There was a legitimate concern that we were going to lose money through that process earlier in the summer. Now, I think we will gain,” he said.

New York state Sen. Brian Kavanagh said New York is in a strong position to gain additional funding.

Photo: Ron Adar/Zuma Press

Landlords and tenants both criticized the state for failing to approve many applications in June and July, and state lawmakers responded in September by extending an eviction moratorium until January.

New York had distributed $18.4 million of rent relief by the end of July, in line with Alabama but far behind Texas and California, which had paid out $893 million and $618 million respectively, according to the latest data from the Treasury Department.

As of this week, New York has distributed more federal rental assistance than any state except Texas, according to data compiled by the National Low Income Housing Coalition, an advocacy group.

New York Gov. Kathy Hochul wrote to Treasury Secretary Janet Yellen on Sept. 21 stating that approximately 9,000 people a week were still applying for assistance, and the state had approved an additional $250 million for landlords and tenants who didn’t meet the income requirements for the federal program.

“Even with these additional state funds, New York’s need far exceeds available funding, and an additional allocation of federal funding will be required to address this crisis,” Ms. Hochul, a Democrat, wrote.

A Treasury Department spokeswoman didn’t return an email seeking comment.

Estimates vary on how many New Yorkers are behind on rent. For the month of August, the Federal Reserve Bank of Philadelphia said some 156,000 New Yorker households were behind on rent due to financial hardships caused by the pandemic, with an average rental debt of $9,200. Some 700,000 New York adults could live in households behind on their rent for any reason, according to U.S. Census surveys.

Jay Martin, executive director of the Community Housing Improvement Program, a landlord group, said it estimates rent arrears in New York are between $2.7 billion and $3.2 billion, on top of the $2.85 billion appropriated by the state and federal governments.

“There is a huge universe of renters who have not applied yet and who are still carrying significant rent debt,” he said. “That is the concern, because when this program runs dry, there will be literally billions of dollars in rent debt that needs to be accounted for.”

Write to Jimmy Vielkind at Jimmy.Vielkind@wsj.com