"Penicillin is required later in the game when The Postal Dude acquires a… umm…well, let’s just say he needs it." — The Gonorrhea medicine's description in the manual
The Gonorrhea Medicine serves to cure the mentioned disease. On Friday, the Postal Dude discovers he has gonorrhea, so he proceeds to go to the Clinic, where Vend-A-Cure gets a piss sample from him. In return, Vend-A-Cure gives him the medicine. Once the Postal Dude takes the medicine, he is cured and can piss normally.
Of note, if the player does not piss during Friday, they will never have to go get the medicine, and Friday's tasks will remain at 3. The only real mechanic of The Dude complaining about pissing is to annoy the player into pissing, thus forcing them to go get the medicine.
- According to the manual, the Gonorrhea medicine is Penicillin.