Delrisa Sewell-Henry, a home health aide, was resigned to spending two hours on a bus and three subway trains just to get across Queens to care for a disabled man.

But when one train recently stopped for a sick passenger, her commute expanded to a mind-boggling three hours and 15 minutes. She was late. To make up the lost time, she stayed longer with her patient instead of picking up her granddaughter from school. She had to enlist a neighbor to do that — for $35.

Then last week, another train was delayed. And she was late again.

“It makes me angry,” said Ms. Sewell-Henry, 54, who earns $13 an hour. “It’s not like we’re sitting around watching T.V. and doing nothing. We’re doing something. We have to be there on time.”

By now, the many ways New York City’s failing subway system has upended countless lives have been well documented. But a new report finds that no group has been hurt more than the city’s army of health care workers who fan out across the city every day to take care of older people, the frail and the sick. They ride trains and buses because they do not earn enough to take taxis or Ubers. Every disruption impacts patients and their families who count on them to arrive on time to provide medication and meals. For the workers, it can lead to reprimands, docked pay and being fired.