From what I recall, the mom left him because he thought the father only wanted the baby and not him from something he overheard him say to his friend. He was depressed and stayed at the friends/band mates house until he ran into baby Byul and the father at a park and they talked and cleared up the misunderstanding.
Oh, his mother left because the father was abusive and was basically just using her for his convenience by selling her to other men. Remember Hyesung didn’t become an omega until the party where he met Dojin where he went into his first heat. Until that point he was an alpha, that’s what he said he was and what his friend thought he was. When his father found out he was an omega, he wanted to use him the same way he used his wife, basically whore him out, so he ran away. That’s when he started staying with his friend first and then later, Dojin.

Didnt the mom abandon him cause he was an omega? I dont remember correctly tho