Staff members at a Bronx public-defenders organization took part in a rap video advocating the killing of police officers and later attempted to mislead city officials about the organization’s involvement, according to the New York City Department of Investigation.

In a report released Thursday, the DOI said two staff attorneys at the Bronx Defenders—a nonprofit group that receives public funds for its advocacy and legal representation for indigent defendants—appeared in a video called “Hands Up,” in which men are shown...