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In this paper we extend and improve all the previous results known inliterature about weighted average, with Ces\`aro weight, of representations ofan integer as sum of a positive arbitrary number of prime powers and anon-negative arbitrary number of squares. Our result includes all cases dealtwith so far and allows us to obtain the best possible outcome using the chosentechnique.

A Ces\`aro average for an additive problem with an arbitrary number of prime powers and squares