Who does Kayn mean when he says "The child"

I think he means himself. Probably more detail in his background lore.

No idea on the lore for the guy but i would assume something along the lines of the nice person inside him (child) was taken over by the bad side (killer).

Probably he is talking about himself, maybe in his past as a child he had to run away and now he instead chases who he wants to kill.

He means himself for sure. It is clear when you see what he says to himself, like "Free the body, kill the body, build a body." He always talking about himself. "Free the body(from Darkin), kill the body (kill that fearfull child inside), build the body (and become an ultimate weapon)." Same is with second quote about mind. Kayn is very deep and... Hmmm... I can say, very dramatic champion. And I think he is one of those psychicaly injured. Maybe he isn't mad like Ziggs or somehing... Bud he is obsessed with power. I dunno how to say it (as English isn't my native language), but... Even as Shadow Assasin he lose against Darkin. He is obssesive and ... And... I cannot find good word. But hope you understand me anyway.

I think that when he says:" The child is gone, killer remains" he means that he's not a child anymore and now it's only the killer inside him

Rhaast: "Yes, he is in your hands."

He Is Talking about Himself After Take The Darkin Blade in another meaning... he's not a child anymore

Read his lore, it'll make sense then