Mike was the cut fresh, wet and actively bleeding? If there is no access to your blood stream then no. If the cut is bleeding then there could be a theoretical risk.
Bear in mind that even in the case of a healthcare worker who should experience and accidental needlestick involving a patient with known hepatitis C the risk of transmission is only about 1.8%.
If concerned get tested with the hep c antibody test after 12 weeks post a concerning exposure then you will know. There is little value at discussing theoretical risk. You and your doctor would be best to judge how risky your exposure may have been.
If the skin was not broken to the blood stream and actively bleeding there is no risk.