  We use recent bounds on bilinear sums with modular square roots to study the
distribution of solutions to congruences $x^2 \equiv p \pmod q$ with primes
$p\le P$ and integers $q \le Q$. This can be considered as a combined scenario
of Duke, Friedlander and Iwaniec with averaging only over the modulus $q$ and
of Dunn, Kerr, Shparlinski and Zaharescu with averaging only over $p$.
