It's hard to believe this even happened, what is with his fingernails, are they like saw blades? And how clean do you suppose his hands were at the time, did he do a medical-style scrub before putting his fingers in your vagina?
Hands have a lot more bacteria on them than vaginas do inside, in fact, the inside of vaginas is (if not fingered) very clean and hands are notable for being very bacterial. It is possible that the cut got infected and that is why it burns.
If you think the cut is really deep, you should see a doctor. you might need a stitch. (Except, it is hard to believe a fingernail could cut you so deeply. Except, you say it burns when you go to the bathroom.) If you think the cut is shallow, take a few sitzbaths (shallow baths) with perhaps some Epsom salts in the water. First wash your area well, then soak in the Epsom-salt water, and when you are out of the bath, put some Neosporin or Polysporin on the cut. If you have the resources, when you are out buying Epsom salts, also buy some wet wipes (get the kind marked "Natural" or some other word, they are in the baby section). Every time you go to the loo for a while, clean the area with a wipe.
About fingering: is it sexually satisfactory for you, or is he just copping a feel? If you don't get much out of having a finger penetrate you deeply (and it is hard to see why you would, all the nerves are in the first third of the vagina and the clitoris is right up top out front), tell Mr. Hands to make a fist, and stick his middle finger (still folded) up about an inch, and to push on your vaginal opening with that instead. It may give you just as much satisfaction as getting penetrated by fingers, especially if your mind starts wandering to the bacterial count on the skin of someone's fingers.
And tell your boyfriend to cut his nails.