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Hello from JustAnswer.
If it is bandaged and clean, then it would generally be OK to drive. The only other issue would be whether there was injury to a tendon, in which case gripping the steering wheel may be difficult. But if you are not have any pain or difficulty with hand grip, then it would be OK to drive.
Even with a small surface cut, if the cut is deep and was squirting blood, it may require repair of the deep structures, but you can drive yourself, if there is adequate hand grip and the wound is bandaged and clean.
If I can provide any further information, please let me know.