A Cesáro 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≤ℓ1≤ℓ2 be two integers, Λ 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≥2 be an integer. We prove that the Ces`aro average of weight k>1 of rℓ1,ℓ2 over the interval [1,N] has a development as a sum of terms depending explicitly on the zeros of the Riemann zeta-function.