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 $n = \phi(n_{1}) + \dots + \phi(n_{j})$, for polynomials $\phi \in \Z[n]$ with $\deg(\phi) = k\ge 1$, $\lead(\phi) = 1$, $j \ge k$, 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 $j=k, k + 1$.