On the average number of representations of an integer as a sum of polynomials computed at prime values

We study the average number of representations of an integer $n$ as $n=\phi (n_1)+\cdots +\phi (n_j)$, for polynomials $\phi \in \mathbb{Z}[n]$ with $\mathrm{\partial }\phi =k\ge 1$, $\mathrm{lead}(\phi )=1$, $j\ge k$, where $n_i$ is a prime power for each $i\in \{1,\dots ,j\}$. We extend the results of Languasco and Zaccagnini (2019), for $k=3$ and $j=4$, and of Cantarini, Gambini and Zaccagnini (2020), where they focused on monomials $\phi (n)=n^k$, $k\ge 2$ and $j=k,k+1$.