This one caused a bit of trouble because it was very similar to another story (called Puppy and Me - I think)  Almost identical in the beginning but this one ended differently. 

Hyuk finds a box with puppy or kitten Hae inside on his doorstep one morning. But as he grew up he fell in love with Hyuk, who was concerned because he was like a father to him, so he tried to discourage it.

Hae had run away from Hyuk because he thought he was unloved. He went to Heechul's house where he got deathly sick because his love was not returned. 

Apparently the same thing had happened with Hangeng when Siwon (his owner) kept him away from his love, Heechul. They called Hyuk to come admit his feelings to save Hae.

Can anyone help?

A big thank you to daisy141 for finding this.

Everlasting Love by Lunar_eclipse (but the account is deactivated)