Most of the remaining homeowners in the program require extensive work. As of this month, the city has finished 189 out of a total of 500 houses that will be rebuilt through the program, or three times more than the 53 houses that were finished a year ago. It has also finished 370 out of a total of 850 homes that will be elevated, or six times more than the 58 elevations a year ago.

Steven Zboinski, 57, whose house on Staten Island is being rebuilt, said the program was finally working. He still remembers submitting required documents in the beginning, only to be told that they had been lost. Soon he was making 10 copies of everything. “It took off to a very bad start and now it’s come back to something that’s actually getting things done,” he said.

But City Councilman Mark Treyger, the chairman of the council’s recovery and resiliency committee, said that despite recent improvements, the Build It Back program overall has been plodding and limited. He said that too many homeowners never enrolled because they did not know about it or had language barriers, while others dropped out after becoming stymied by conflicting and confusing information. As a result, he said, many have depleted their savings, remain stuck in temporary housing and face a desperate future.

“Yes, progress has been made but it should not be celebrated because we have a long way to go,” said Mr. Treyger, a Democrat who represents storm-battered Coney Island, Sea Gate, and Gravesend in Brooklyn. He has called for the city to reopen the program to more applicants.