Congratulations, you won the New York City housing lottery! But not so fast: Before you can move into an affordable apartment, you have to provide a mountain of paperwork to verify you’re eligible.

Just 151 people have left state incarceration out of hundreds potentially eligible as state grapples with a shortage of correction officers. Housing restrictions are a big reason why.

In response to a relentless rise in calls to New York state’s child abuse hotline, alongside stark racial disparities, social ...