Homer gives himself a cut on his forehead in the shower, shortly afterward he is at dinner with Hap and numerous closeups of his face show no cut. Later when he is back in the hotel room, the cut appears again on his face when Renata asks him how he got it.
The imprisoned characters' hair never changes after years in captivity. Homer's especially would need to be cut regularly to maintain its short length, but how that possibly could, or would happen, is a mystery.