I thought I would share this question as it might be interesting for other users here.
Assume that a function which is in a uniform class (like $NP$) is also in a small nonuniform class (like $AC^0/poly$, i.e. nonuniform $AC^0$), does this imply that the function is contained in a smaller uniform class (like $P$)? If the answer to this question is positive, what is the smallest uniform complexity class that contains $NP \cap AC^0/poly$? If negative, can we find an interesting natural counterexample?
Is $AC^0/poly \cap NP$ contained in $P$?
Note: A friend has already partially answered my question offline, I will add his answer if he doesn't add it himself.
The question is my second attempt to formalize the following informal question:
Can non-uniformity help us in computing natural uniform problems?