How Should A Finger Cut With Scabbing And Pus Be Treated?

hi, ummmm, i have this small cut from picking a hangnail on my finger, it seemed to heal, but the then the scab turned greenish,and when i pick at it and cleaned it today, greenish like puss came out, not too much ,but enough to frighten me, i am putting peroxide and neosorin on it, what do you recommend?