From the description, the bump seems to be due to infection which you might have caught either through the sexual act or by razor while shaving. The infections usually does not have pus from the beginning. Initially, there is swelling of the tissue what we call as
cellulitis which is painful bump. When this is tried to open, blood comes out. If treated well in time and with proper antibiotics, it regresses. If left untreated then the infected cells start getting damaged and pus is formed. This stage is called as
abscess. In this stage, even the antibiotics will not cure the infection. When abscess is formed, it has to be incised and the pus has to be drained.
From your history, you seem to have infection which is at the cellulitis stage. Better to start the antibiotics before it progresses to the abscess formation.
But do not take antibiotics of your own. Consult your Gynae or Skin & VD physician and on his/her advice only start the proper antibiotics.
There is a small chance that on actual clinical examination, the bump may turn out to be something else. However, with whatever information you have provided, it seems to be infection.
One more advice, while taking the treatment for yourself, please get your sexual partner also checked for any source of infection and if so, get him also treated to avoid the repeat infections.