Ms. Muñiz stayed in the room with her husband while he received radiation treatment. “I’m willing to take anything with him,” she said. “I never left him alone, and I never will.”

Radiation did not work. To save Mr. Muñiz’s life, surgeons removed his tongue and his lower jaw and cut a hole through his esophagus. Disfigured, depressed and unable to speak, he can consume nothing thicker than milk and needs near-constant care.

This is the household where the Muñiz children grew up.

“We’ve been through every craziness,” said José Jr., 24, who has suffered depression so severe that he dropped out of college and confined himself to the apartment, “every up and down.”

For years at a time, the family held on, seemingly by a thread.

Over the summer, the younger son, Jesus, 22, got a part-time job at a Zaro’s Bakery in Manhattan’s financial district. Because the family’s rent is tied to income, the rent tripled in August, to about $770 a month from $245. But Jesus had school bills to pay, and the family paid some of the funeral expenses for José Sr.’s mother, who died over the summer, and things began to unravel.

They fell behind on the rent and utilities. Food was often scarce. The family regularly skipped meals.

It was around this time that Ms. Muñiz got in touch with Catholic Charities Archdiocese of New York, one of the eight organizations supported by The New York Times’s Neediest Cases Fund. It covered their back rent, got them warm coats and blankets and helped them apply for food stamps for the first time. And with $600 from the Neediest Cases Fund, the family paid its electric bill.