Find all functions such that $f(m)+f(n)|m^p+n^p$

Question

For fixed prime number $p$, find all $f:\mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ such that $f(m)+f(n)\mid m^p+n^p$ for all $m,n\in \mathbb{Z}^+$

I managed to get only that for prime $q$ we have $f(q)=q^k$ for some nonnegative integer $k$.