On the Ces\`aro average of the "Linnik numbers"

Let $\mathrm{\Lambda }$ be the von Mangoldt function and $r_Q\left(n\right)={\sum }_{m_1+m_2^2+m_3^2=n}\mathrm{\Lambda }\left(m_1\right)$ be the counting function for the numbers that can be written as sum of a prime and two squares (that we will call Linnik numbers, for brevity). Let $N$ a sufficiently large integer, let $k>3\mathrm{/}2$ and let $M_{i}\left(N,k\right),, i=1,\dots,k$suitableparametersdependingon$J_{v}\left(u\right)$,where$J_{v}\left(u\right)$denotestheBesselfunctionofcomplexorder$v$ and real argument $u$. We prove that $$\sum _{n\le N}{r}_Q\left(n\right)\frac{{(N-n)}^k}{\mathrm{\Gamma }(k+1)}=M_1(N,k)+M_2(N,k)+M_3(N,k)+M_4(N,k)+O\left(N^{k+1}\right)\mathrm{.}$$ We also prove that with this technique the bound $k>3\mathrm{/}2$ is optimal.