A Ces\`aro average for an additive problem with prime powers

In this paper we extend and improve our results on weighted averages for the number of representations of an integer as a sum of two powers of primes. Let $1\le \ell_1 \le \ell_2$ be two integers, $\Lambda$ be the von Mangoldt function and % $r_{\ell_1,\ell_2}(n) = \sum_{m_1^{\ell_1} + m_2^{\ell_2}= n} \Lambda(m_1) \Lambda(m_2)$ % be the weighted counting function for the number of representation of an integer as a sum of two prime powers. Let $N \geq 2$ be an integer. We prove that the Ces\`aro average of weight $k > 1$ of $r_{\ell_1,\ell_2}$ over the interval $[1, N]$ has a development as a sum of terms depending explicitly on the zeros of the Riemann zeta-function.