I once cleaned up an infected cut myself, but it was very minor and it was not deep. The pus continued for a while and then it stopped. Antibiotic cream really made a difference.
The one thing I paid attention to was that I didn't apply any other lotion on the cut because it was on my face. I applied regular lotion to the rest of my face, but not the cut because I tried it one day and it made things worse. The cut became swollen.
After that I tried to keep the cut as dry as possible and only used a small amount antibiotic cream twice a day. That seemed to do the trick and it healed within a week.