I recommend the Durihana Children and Adolescent Clinic in Sanseong-dong, Daejeon.

This is a place I've been going to since I was a child. First of all, the teachers are friendly. The nurses are also kind and helpful, and they guide us quickly. Since the center is for children and adolescents, the teachers are also very understanding. They seem to explain things well, both from a child's perspective and to parents! Best of all, the ratings are quite high! ^^