How powerful is $ACC^0$ circuit class in average case?
There seem to be typos in what you wrote; I take it you're asking: is there a function $f \in NEXP$ such that for all polynomials $p$ and infinitely many $n$, no ACC0 circuit of $p(n)$ size can compute $f_n$ on more than $1-1/p(n)$ inputs of length $n$?

As far as I know, this is open. But here is a possible path to doing it. We know that every $NEXP$-complete function $f$ does not have ACC0 circuits of polynomial size. Perhaps an arithmetization of some $f$ (representing $f$ naturally as a family of multilinear polynomials) over a large field of finite characteristic (e.g. characteristic two) would work. (Arithmetizations can typically give you average-case hardness "for free" ... but one would have to carry out the finite field arithmetic in ACC0, presumably.) Cite me if you figure it out. ;)