I think because hes in some sort of competition with naeun, where has bas like a superiority complex and wants to make her seem somewhat vulnerable, he wants to do a favor for her or smth so she feels indebted.. im not sure. I also think he was treating garam like a fling like he does with everyone else, expecting him to be ruined by him like everyone else, but be developed feelings for him at some point. Thats why he called off things with yonseop or whatever his name is.
his mission was to make naeun owe him a favor, due to her relationship with the chairwoman and what he wanted being something only the chairwoman had the power to make happen. the favor was for naeun to ask the chairwoman to reinstate a man from his church with his former position as “first violin” in the gaeum philharmonic orchestra.
the reason the man had to resign from that position was because people found out he was having an affair with a student (i.e., youngjo). this added to youngjo’s complex about how he thinks he ruins the lives of people who love him, so youngjo wanted to give back the man his position in order to feel better about himself. unfortunately for youngjo, the man ended up not even wanting the position back due to his reputation being completely ruined, which he still blames youngjo for. so, the entire plan was meaningless in the end.
oh, also youngjo used garam in order to get naeum to owe him a favor because her only interest is the piano and she saw garam as her competition. youngjo gambled on his belief that she would want to sabotage garam, and she did.
so, youngjo planned to “ruin” garam like he did to everyone who loved him before, and in order to do that he needed garam to fall in love with him first.
however, garam wasn’t opening up as much or as quickly as youngjo needed to make that happen, which meant he needed garam to trust him even more. so, he spread the rumors to put garam in an even worse place mentally and emotionally, which gave him an opportunity to “save” garam by keeping him company at all times while he was being bullied. it worked because him and garam got much closer, which put his plan back on track.

It's so goated. I'm glad I found it.
By the way, can someone explain to me what Youngjo real mission was? I'm so confused. He didn't want to hurt Garam, but why did he start the rumour and handshake with naeun?
I would be lying if I said I didn't like him, but I'm also happy that Garam didn't let him go all the way. His situation is sad, but nonetheless, he is still a playboy bastard.