Quickest way to balance the public housing budget: abolish public housing. Evict everybody. Bulldoze the buildings. Sell the land to private developers to build condos
This has been tried with heavily subsidized buyouts. The tennents rioted because they would lose thier handout. Frankly, no 'pour' person should have a car in the city. Those spaces could be used by commuters and the city would over $200 per month per space in manhattan.
I wouldn't say sell it to folks to build condos, I'd open it up to anyone who wants to pay the most for the land, period.
Public Housing in NYC was created by the Gov't of NYC because the government of NYC created a housing problem on its own.
I.E. The government did something stupid which reduced supply and the incentive to build or develop more of a supply, so the governmnt in its infinitate stupidity, had to create public housing which is a human hell hole to fix the problem it created, by making a new problem and then making it even worse since it can't do even the most basic things right.