Mayor Bill de Blasio’s administration frequently says the number of homeless in New York City is about 60,000 people sleeping in shelters and 3,675 living on the streets.

However, the U.S. Department of Housing and Urban Development says it will release a report next month that puts the population at 78,604. Advocates for the homeless who have seen HUD’s count in recent weeks say it adds to the urgency for the mayor to build thousands of units of permanent affordable housing. They also accuse the city of inaccurately portraying...