Mr. Cuomo has also pointed to the subway’s rising homeless population as part of an increase in “quality of life issues.”

There were 2,178 homeless people on the subway during one day in January this year, compared to 1,771 last year — a more than 20 percent jump, according to a report by the transit agency. The report recommends that the new M.T.A. officers enforce rules about not sleeping or panhandling on the subway and help with outreach to access shelters.

Expanding the police force will likely add to the transit agency’s financial challenges.

The annual cost for one new officer is nearly $93,000 in the first year, including overtime, health benefits and pension costs, according to a report by the Citizens Budget Commission. Hiring 500 officers and 81 supervisors will cost $56 million in the first year and grow to nearly $120 million per year in a decade.

The authority has a nearly $17 billion annual operating budget, and officials expect to have huge deficits in the coming years, reaching $433 million by 2023. But the agency’s separate capital budget is expected to get an influx of cash after Mr. Cuomo convinced state lawmakers to approve congestion pricing, a plan to toll drivers entering Manhattan that could raise $15 billion.

At the same time, the subway is struggling to rebound. The on-time rate for trains recently rose above 80 percent for the first time in years, from 65 percent in 2017. But there were still more than 35,000 train delays in September, including a train that broke down in Brooklyn in a smoke-filled tunnel.

Transit officials are also negotiating a new union contract with Transport Workers Union Local 100, which represents more than 38,000 subway and bus workers. The negotiations have turned nasty, and workers recently held a large rally outside M.T.A. headquarters.

The union, which has been a close ally of Mr. Cuomo’s and a major contributor to his campaigns, has won lucrative contracts in recent years. But major wage increases for workers could deepen the authority’s budget hole.