On the Gauss circle problem over smooth numbers

The Gauss circle problem concerns with the evaluation of ${\sum }_{n\le x}r(n)$, where $r(n)$ denotes the number of representations of $n$ as sums of two squares and $x\ge 2$. Let ${\mathrm{\Psi }}_G(x,y)$ denote the sum of $y$-smooth numbers below $x$ weighted by $r(n)$. In this paper, we evaluate ${\mathrm{\Psi }}_G(x,y)$ asymptotically for certain ranges of $x\ge y\ge 2$.