The Cuckoo

Patience Bellamy is a middle- aged spinster who lives an unremarkable life trying to make ends meet until, one day, she comes home to a basket and several packages on her doorstep. Improbably, in the basket is an infant, wrapped in a blue blanket. The note pinned to the blanket explains that the parents can no longer care for the child. They implore Patience to provide a home for their child, and they futher warn her against taking the child to the doctor's office. And, the note adds, she'll understand once she attempts to feed the baby.
This novella has also been collected in the anthology "Brushes with the Supernatural."