Main threat is rabies, which your pet should not have. Unless you need stitches, a good rinse with something sterile and a bandage are all you need. That’s all a hospital is going to give you. Maybe some antibiotics if it didn’t get cleaned quickly enough.
If it’s your own cat and you do regular vet visits, it’s fine as long as you clean the cuts properly.
Not sure what use vet visits are here? Why would I risk anything if my hand has been bit deep enough to draw blood, even if it’s my own cat?
Main threat is rabies, which your pet should not have. Unless you need stitches, a good rinse with something sterile and a bandage are all you need. That’s all a hospital is going to give you. Maybe some antibiotics if it didn’t get cleaned quickly enough.