New York City officials recently gave $5.3 million in contracts to a nonprofit that owes the state $143,678 for allegedly misusing grant money.

The city’s Department for the Aging awarded the contracts to the nonprofit Institute for the Puerto Rican/Hispanic Elderly last month to provide services for senior citizens in the Bronx, Queens and Manhattan between Dec. 1, 2019, and June 30, 2021.

This...