My Kid Injured Their Finger About A Month Ago Playing.

My kid injured their finger about a month ago playing. It became infected and was treated with bactine and antibiotics. A couple weeks later doctors diagnosed him with impetigo for a small rash and they blamed the finger which looked worse than I remembered. Now the finger is still sensitive. Has no cuticle and it's peeling from side and cuticle... can you give me some sort of heads up as to what this might be.... unfortunately I feel that the doctor has done a bit of misdiagnosing.