Non-Uniform vs. Uniform Adversaries
There is hardly any natural problem which is believed to be in P/poly but not in P. The artificial examples can be adapted to answer your question.

Assume $E \neq NE$, then there is a unary language L in NP which is not in P -- unary means that all the strings in the language have the form $1^n$ for some n.

Then define L' to be the set of all strings x such that $1^{length(x)}$ is in L. This is in NP, it is in P/poly, and it is not in average polynomial time