Children with illnesses are often seen as undesirable and costly in China, driving many families to abort or abandon them. Abandoning a minor is a crime in China punishable by up to five years in prison, and courts have argued that if a child dies after being left in dangerous conditions, it is equivalent to murder.

“Parents are often nice people who feel hopeless raising these children themselves and believe they will have a better life with a better family,” said Qiu Lili, founder of Beijing Angel Mom Charity Foundation, which helps care for abandoned children.

A health care system marked by high co-payments and deductibles, as well as inconsistent coverage, means that paying for treatments can be devastating for low-income families. Adding to the problem, many families lack insurance coverage for infants, advocates say.

Ms. Qiu said that while families with disabled children in large cities like Beijing and Shanghai receive generous subsidies, those living in smaller cities often lack the resources to give them long-term care.

“Child welfare has improved recently in China, but it’s still not good enough,” she said.

The parents of the 6-month-old girl said in their letter that they had already depleted their savings, spending 60,000 renminbi, or about $9,400, on treatment.

“There’s no money to help her recover and she can’t take care of herself,” the letter said. “Her life is doomed.”