Refuah sheleimah!
You should gently clean and soak your hand before mikveh. If there is a piece of hanging skin that is clearly attached only on one side, you can trim it (doctor permitting). However, we do not recommend pulling off hanging skin, since it is likely to lead to more skin peeling off.
Please also check in with your physician regarding concerns about infection. If there is a concern, you can check with the doctor and the mikveh attendant about arranging to be the first to immerse on your mikveh night.