Let's define for a fixed odd prime power $p^m$, $$r_{p^m,k}(n)=\#\{(a_1,\dots,a_k)\in\mathbb{Z}^k:\sum_{i=1}^ka_i^2=n,p^m|n\}.$$ How can we find an expression and growth bound for $r_{p^m,k}(n)$ as we have for $r_k(n) =r_{1,k}(n)$ using singular series? I am interested only in the case of $k=4$ but would be happy to know in general?

Any reference will be highly helpful.