When you say cut, I have heard the word "fresh cut" - what does this mean? Can you get infected through "nicks" in the skin that do not bleed?
The only data I can refer to in responding to your question involve a case of a mother with severe excema on her hands who cared at home for her son with AIDS, and didn't wear gloves in contact with body fluids. She became infected, presumably from infected blood making contact with the weeping excema on her hands. Beyond that, I am not aware of infection through "nicks." The key is access to the bloodstream. A partially healed cut would offer more protection than severe excema, but all of these things are small points on a scale.