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 $1\le {\mathrm{\ell }}_1\le {\mathrm{\ell }}_2$ be two integers, $\mathrm{\Lambda }$ 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 $N\ge 2$ be an integer. We prove that the Ces`aro average of weight $k>1$ of $r_{{\mathrm{\ell }}_1,{\mathrm{\ell }}_2}$ over the interval $[1,N]$ has a development as a sum of terms depending explicitly on the zeros of the Riemann zeta-function.