I would definitely flush the wounds with chlorhexidine(diluted to normal ratios), and maybe some warm water afterwards. If you have some povidone or betadine, that would be fine too. Let dry, then if you have a polysporin ointment, rub a bit of that in to the wounds. Cat claws are gross as are their mouths. I really don't know what the vet would do, other than tell you to scrub the wounds with a chlorhexidine soap or betadine, put polysporin on it and try to keep the outside of the wounds open so they can heal inside out. I don't think a round of Abx would be necessary if you washed the wounds fairly quickly, but the vet may say otherwise.

Poor booger! Give that cat a squirt with the spray bottle for me