It's not just that, but Kye Sook is a shady character. We don't know anything of his background, we don't know of his status with Soo-Won. All we know is that he's an advisor. I believe he may be trying to gain the throne for himself, not for Soo-Won's cause. Notice how he tries to kill Yona quickly when they see each other again. She even states that she needs to see him, but he still wants to kill her anyways. He wanted to kill her in the first place to keep her quiet. However, there could also be another reason, which could be that she may try and sway Soo-Won's decisions. She could overpower Kye Sook on demands and suggestions, and that may be why he wanted to kill her in the first place. He's been a shady character since the beginning, and it may only be a matter of time before he turns on Soo-Won.
Also, he seems to have some control over Soo-Won. In the second chapter, he was only going to kill Yona because Kye Sook recommended it. It didn't cross his mind to kill her until that.

I hate that Kye Sook guy.
While I'm not super okay with Soo Won, at least he s trying to do sth good, he feels bad for Yona and we saw that he really loved her.
But that guy comes out of nowhere, doesn't listen to her and just ask to kill her like that?
Srsly, fuck you, I dont usually swear but you deserve it...
You show him, girl, show them all how you're so awesome and how they can't mistreat you and should listen to you.
Go all badass on them and SHOW that kye Sook guy what you're made off!!