More than a quarter of the teachers have been in the pool for at least five years, the report stated. About a third of the teachers are there because of a past legal or disciplinary case against them, according to the latest statistics provided by the city.

Mayor de Blasio vowed to shrink the pool, and his efforts have had an impact. The number of teachers in the pool at the start of the 2017-18 school year, compared to four years earlier, was down by 39 percent, the city said. In the 2016-17 school year, the pool cost the city $152 million, $16 million more than this year.

In an effort to get that number down further, the city last year offered financial incentives for schools to hire teachers from the pool and said that it would place as many as 400 teachers in vacancies, whether principals wanted them or not. They would have a year to prove their abilities, after which the city would consider taking steps to dismiss them if they didn’t measure up.

But according to the report, only 75 teachers actually got placed because, the city said, there were fewer vacancies than expected. The city said 372 more teachers were hired by schools under the incentive program. And an additional 170 teachers in the pool took buyouts earlier this year, saving the city about $23 million.

Will Mantell, a spokesman for the education department, said, “We’ve made common-sense reforms to the Absent Teacher Reserve, cut the pool down by 450 teachers in two years and are on track to reduce the pool by half for next year.”

Independent studies have raised concerns over where these teachers ended up. Most of the reserve pool teachers were placed in low-performing schools with higher than average populations of black students, according to an analysis done by the Education Trust-New York.

Some cities have time limits for unassigned teachers. In Chicago, teachers without placements have 10 months to search for a new position, and are then let go. In Washington D.C., they have 60 days to find a new job. After that, those with unsatisfactory performance reviews can be let go, while those with satisfactory reviews can either take a $25,000 buyout, retire or spend one more year looking for a job, the report stated.

Maria Doulis, vice president of the Citizens Budget Commission, agreed that Mr. de Blasio’s efforts to shrink the pool have had some effect. “They’ve been trying, and there is some modest movement,” she said. “But the trajectory is that more and more teachers and other personnel are ending up in the ATR, so it’s just a constant hustle and grind to get these folks placed. The question is: How long should we subsidize their employment?”