Refined Hardy–Littlewood conjecture in arithmetic progressions

For $k\geq1$, let $\mathcal{H}=\{h_1,\dots,h_k\}\subseteq\mathbb{Z}$, let $q\equiv1\,(\mathrm{mod}\,4)$ be prime, let $a\in\mathbb{Z}$, and let $R_k(\mathcal{H};x,q,a)$ and $\mathfrak{S}(\mathcal{H})$ be as above. Let $c_{1,h}$ be the residue-class secondary constant defined in the source. Refined Hardy–Littlewood conjecture. If $\mathfrak{S}(\mathcal{H})>0$, then

$$R_k(\mathcal{H};x,q,a)\sim\frac{\mathfrak{S}(\mathcal{H})}{q}K^k\left(\frac{1}{(\log x)^{k/2}}+\frac{1}{(\log x)^{k/2+1}}\sum_{h\in\mathcal{H}}c_{1,h+a}+O\left(\frac{1}{(\log x)^{k/2+2}}\right)\right).$$

This refines the preceding arithmetic-progression heuristic by incorporating secondary terms for individual sums of two squares; it remains open.