On Wednesday, New York City hit a record 59,373 people in shelters overseen by the Department of Homeless Services.

There is no clearer indicator of the homelessness crisis than in the Bronx at the intake center for families with children, where on a recent Saturday morning at 7:30, Larissa Galindo had just gotten off the bus from a temporary shelter.

“We’re tired,” Ms. Galindo, 19, said, burying her face in her hands and trying to wipe the sleep and frustration from her eyes. Unique, her 1-year-old daughter, looked up from a stroller. They had left the center, known as PATH, short for Prevention Assistance and Temporary Housing, four hours earlier.

“I want to go to sleep,” Ms. Galindo said.

In a cascade of good intentions and unintended consequences, homeless parents and their children are facing dayslong waits and sleepless nights as they flood the city’s already overwhelmed homeless services.