The average value of a certain number-theoretic function over the primes

We consider functions $F:\mathbb{Z}_{\geq 0}\rightarrow\mathbb{Z}_{\geq 0}$ for which there exists a positive integer $n$ such that two conditions hold: $F(p)$ divides $n$ for every prime $p$, and for each divisor $d$ of $n$ and every prime $p$, we have that $d$ divides $F(p)$ iff $d$ divides $F(p \mod d)$. Following an approach of Khrennikov and Nilsson, we employ the prime number theorem for arithmetic progressions to derive an expression for the average value of such an $F$ over all primes $p$, recovering a theorem of these authors as a special case. As an application, we compute the average number of $r$-periodic points of a multivariate power map defined on a product $Z_{f_1(p)}\times\cdots\times Z_{f_m(p)}$ of cyclic groups, where $f_i(t)$ is a polynomial.