If it’s an infection don’t take anti-inflammation drugs, because something beeing inflamed is just the immune system doing it’s job. Unless you know what you are doing (like a doctor) you definitely shouldn’t shut off your immune system.
What you can do however is apply a fat based antibiotikum, like mosts salves or oils. My personal, scientifically not really verifyed, recommendation is to use lavernder oil, it has antibacterial and fungal properties, accelerates the first stage of wound healing (scientifically verified at least for mice), however not the overall duration. It’s also fat based, so it can dissolve into the skin tissue and the wound.