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For a positive integer $d$, let $h(-d)$ denote the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-d})$. Is there a known asymptotic formula for the sum

$$\displaystyle \sum_{\substack{p \leq x \\ p \equiv 3 \pmod{4}}} h(-p)?$$

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It seems to me one should be able to derive such a formula from the class number formula and the known statistical results on $L(1,\chi)$. But... I see now this has already been worked out. See

Nagoshi, Hirofumi(J-GUN-FS) The moments and statistical distribution of class numbers of quadratic fields with prime discriminant. (English summary) Lith. Math. J. 52 (2012), no. 1, 77–94.

According to the MathSciNet review, the author gets asymptotic formulas for all the (positive integral) moments of $h(-p)$, for $p \equiv 3\pmod{4}$.

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