Steve didn't wanna kill the Winter Solider. Yes, he was his mission but his mission was to bring the Winter Solider to justice, to imprison him. Steve decided differently once he realized who that solider was.
"From the moment Odin saw Loki, he saw an instrument to end a war. From the moment Loki saw Odin, he saw someone he could love. Loki wanted Odin's approval so much he changed his form to look like an Asgardian. Even as a newborn.