White Valentine (1999)

A girl named Jung-min writes to a young man named Hyun-jun, who is serving in the army. She lies about her age and claims to be a teacher. As Jung-min turns twenty, a 30-year-old man, who has eyes filled with sadness moves into her village. Every night, he sends letters through a pigeon to a deceased woman he loved dearly. He sends them high up in the sky, knowing that he will never get a reply. Then one day, like magic, he receives a letter. The letters sent through pigeons reveal someone's loneliness and sadness. And the two people meet each other by sheer chance..

Unfortunately the movie White Valentine is not yet available on HBO Max.