Almost 20,000 children now go to bed every night in New York City’s homeless shelters, up from 16,000 last year. If homelessness is hard on adults, for the young, it can be disastrous, starting a slide into a lifetime of problems. Mayor Michael Bloomberg and Gov. Andrew Cuomo could reduce these numbers by restarting programs that help homeless families move from shelters to permanent rental housing. In 2005, Mr. Bloomberg promised that he would cut the city’s homeless population of 33,000 by two-thirds by 2009. It hasn’t happened; the total homeless population has reached almost 46,000 this year. He often blames the economy, which is certainly a factor in rising homelessness. But he and the governor have eliminated important programs that helped shift people out of shelters. It is time to restore the cuts and reinstate those programs. The Bloomberg administration unwisely ended priority referrals for homeless families to public housing and for federal rent subsidies, which have very long waiting lists. The mayor should find a way to give destitute families quicker access to public housing and rental vouchers.