Yes, especially if the cut is somewhere that it's at risk of being damaged again. For example, if I get cut on my fingers while at work, it's likely that that finger will endure more punishment as the night goes on. I'm no connoisseur of pain, and I don't feel like trying to scrub dirt out of an open cut repeatedly. Keep a bandage over it, and after a few days it's sufficiently healed.
I prefer the fabric bandages, too. If I use the plastic ones (even the latex-free ones), I develop a blistery rash where the thing adheres to my skin. The fabric ones also have better flexibility and adhesion.