Bounding the exponential sum on squares of some sifted sequences

Let $\mathfrak{B}$ denote the collection of odd primitive Gaussian integers and $n\mapsto b(n)$ denote the characteristic function of elements of $\mathfrak{B}$. We prove that the exponential sum $S(\alpha ;N)={\sum }_{n\le N}b(n)e(n^2\alpha )$ satisfies \begin{equation*} \frac{S(α;N)}{N/\sqrt{\log N}} \ll N^ε(q^{-1/4}+N^{-1/2}q^{1/4}+N^{-1/8}), \end{equation*} where, $(a,q)=1$ and $\mathrm{\mid }\alpha -a\mathrm{/}q\mathrm{\mid }<1\mathrm{/}{q}^2$. Though we specialized on sums of two squares, these results extend to more general sequences.