>>2926847She absorbed an Angel and her left eye was heavily corrupted, altering its appearance. The corruption being so great there, a sealing pillar was implanted in it to weaken the corruption (even though it had affected her entire body). The corruption caused Asuka to lose the need/freedom to sleep and eat.
There's no practical explanation given for the patch. For Asuka, if I had to guess, it's probably simple. She hates to look at it. She hates the curse of the Eva (being unable to age) and now also the curse of the Angel she absorbed. Her eye marking a clear a brand of that curse. She hates when others can see it.
