Solutions of $x_1^2+x_2^2-x_3^2=n^2$ with small $x_3$, revisited

Let $r(N)$ be the number of representations of $N$ as a sum of two squares. For every integer $n\ge 2$ we prove an asymptotic formula for $\sum_{x\equiv e \bmod 2}Φ(x/M)\,r(n^2+x^2)$, that is, for the number of integral solutions of $x_1^2+x_2^2-x_3^2=n^2$ with $x_3\equiv e \bmod 2$ in a range $|x_3