Thanks for writing your query.
Your problem seems to be due to infection due to ingrown toe nail.
As the swelling has bursted and it is bleeding along with pus discharge, you should consult a surgeon for proper incision and drainage
of the swelling. You should start oral antibiotics
and anti-inflammatory medicines after consulting your doctor.
Continue applying Neosporin
You can also have a Culture and Senstivity test of the pus draining from the wound, as this will help to know the causative organism and the appropiate antibiotic needed.
I hope this is helpful to you.