So what is it? Why is it there? Rudy's definitely not old enough to shave, so that's probably not a razor nick. Even if it WAS, it wouldn't last for the whole darned game. Maybe he's really lazy and just never bothered to take off the bandage. But given the average useful life of a Band-Aid, I tend to doubt that any bandage, except possibly one made entirely out of duct tape, could remain on Rudy's face for three or four very active months. Even if it did so, it wouldn't remain that pristine white color.

Perhaps it's not a bandage at all, but some sort of simple yet complex tribal marking! Perhaps the white marking on his left cheek means he's a junior member of some aboriginal tribe! Uh... no. Never mind.

If Cecilia or Calamity Jane were giving him hickies, they'd be more likely to be on his neck. If Rudy ever shows up with TWO bandages, one on his neck, I'll personally laugh my head off. Plus he had the bandage before he met either of those girls, and I don't think any of the Surf Village girls were giving him permanent cheek-hickies. (Poor Rudy.)

I suppose it's possible that Rudy thinks that wearing the bandage gives him a certain rakish and slightly-dangerous air. That seems like something that Jack might do, certainly, but Rudy is fairly un-self-centered. Anyone who wears that goony armored thing on his left arm can't be THAT fashion-conscious.

Sooo... all spurious theorizing aside... why IS there a bandage on Rudy's cheek?

Here's my hypothesis.

At one particularly fraught moment in the game, Rudy (urk) cuts his own forearm off to get away from the villain's entangling whip, an act which ends up saving the lives of all three characters. However, it also reveals to the other two characters and the world that Rudy is, in fact, a robot. A hideously complicated war-bot, actually, which is why he was able to bond with ARMs so easily. So, anyway, they manage to save his life after a bunch of complex maneuvers, part of which involves forging him a new arm (no, not a new ARM, you comedians).

So, Rudy's a robot. Furthermore, the technology that produced him is no longer accessible except in highly dangerous places. Therefore, to my hypothesis.

At one point in his life, Rudy suffered a wound to his left cheek that was fairly deep. As a result of this wound, the circuitry/metal armature underneath his 'skin' was revealed, and the skin was removed to such an extent that it couldn't be patched or healed. Perhaps it was a very bad burn, or perhaps his grandfather needed to radically repair something in his face that could only be reached that way.

Not having the equipment necessary to repair himself, due to the lack of comparable technology, Rudy's faced with a choice. He can either walk around with gleaming metal shining out of his cheek, proclaiming his inhuman origins to the world, or... he can wear a bandage! Alternately, the bandage may prevent oil and other fluids from leaking from his endoskeleton, in essence holding him together.

So, you ask, why can't Cecilia heal it? Maybe she can, and Rudy's just gotten used to the bandage. (We do know that Cecilia's Crest Graph magic works normally on Rudy.) Or maybe Crest Graph healing only works on fairly recent wounds. Perhaps there's something odd about that particular wound which makes it resistant to magic. After all, she couldn't just heal him a new arm.

Now, then, imagine that Rudy didn't know he was a war-bot. (It's been a while since I've played, so I can't remember if he knew or not.) And after hurting himself badly, he looks at himself in a pool of water and sees... gleaming metal in his cheek. Once he discovers that it's part of him and not a foreign object, he absolutely freaks out and covers it up the best he can, never letting himself see it, banishing the thought of it, as best he can, from his mind. In total denial, even after the others discover his secret, he continues to bandage the place, never knowing if Cecilia managed to heal it or not, because he refuses to look and see. Even after he and the others have come to terms with his nature, he would still bandage the place as a sop to his old, buried phobia.

In short, the 'bandage' is either a clever ruse to disguise Rudy's obvious metallic nature, revealed by a wound in his jaw, or it helps Rudy maintain his sanity. It may serve a lesser purpose as well, but those are, in essence, my hypotheses.