To the Last Day (1960)

Captain Kim is grievously wounded in the first days of the war. When the northern troops continue their advance, his wife and small daughters flee south to Taegu, dragging him along in a two-wheel handcart. His wife struggles to keep herself and their surviving daughter alive. Working as a market trader, she meets a kind young man -- who happens to be tall, dark and handsome.

Unfortunately the movie To the Last Day is not yet available on HBO Max.