You did me wrong.
In that sentence, is wrong an adverb or some other part of speech? I don’t understand the syntactic construction being used here.
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Like Toothrot, I would consider this use of wrong to be a noun. For at least some speakers, it can be negated with no, as "You did me no wrong" (e.g. in some translations of Galatians 4:12). If wrong were an adjective or adverb in this sentence, we wouldn't expect this kind of negation to be possible.
Similarly structured expressions (although they have different meanings) are "to do one good" and "to do one harm".
There was a previous question asking about why people say "to do one wrong": The grammatical strangeness of "done me wrong" and "did me service"
wrong is here a noun, or, if you will, an adjective used as a noun.
You did (unto) me (a) wrong.