Sums of Hurwitz class numbers, CM modular forms, and primes of the form $x^2+n{y}^2$

We consider sums of Hurwitz class numbers of the type ${\sum }_{t\equiv m(\mathrm{m}\mathrm{o}\mathrm{d}M)}H(4p-t^2)$, where $M$ is composite. For $M=6$ and $8$, we show that these sums can be expressed in terms of coefficients of CM cusp forms. This leads to explicit formulas which depend on the expression of $p$ in the form $x^2+n{y}^2$.