The trigonometric polynomial on sums of two squares, an additive problem and generalisation

Let $B$ be the set of odd integers that are sums of two coprime squares. We prove that the trigonometric polynomial $S(\alpha;N)=\sum_{b\in B,b\leq N} e(b\alpha)$ satisfies $$\frac{S(\alpha; N)}{N/\sqrt{\log N}}<<_{A,A'} \frac{1}{\phi(q)} + \sqrt{\frac{q}{N}}(\log N)^{7} +\frac{1}{(\log N)^A}$$ for any $A,A'\geq 0$ and when $(a,q)=1$ and $|q\alpha-a|\leq (\log N)^{A'}/N$. We use this estimate together with a variant of the circle method influenced by Green and Tao's Transference Principle to obtain the number of representations of a large enough odd integer $N$ as a sum $b+b_1+b_2$, where $b\in B$ while $b_1$ (resp. $b_2$) belongs to a general subset $B_1$ (resp. $B_2$) of $B$ of relative positive density. We further show that the above bound is effective when $0\leq A<1/2$.