In this paper, we improve the moment estimates for the gaps between numbers that can be represented as a sum of two squares of integers. We consider a certain sum of Bessel functions and prove the upper bound for its mean value. This bound provides estimates for the
$\unicode[STIX]{x1D6FE}$
th moments of gaps for all
$\unicode[STIX]{x1D6FE}\leqslant 2$
.