The multiplication table constant and sums of two squares

We will show that the number of integers $\leq x$ that can be written as the square of an integer plus the square of a prime equals $\frac{\pi}{2} \cdot \frac{x}{\log x}$ minus a secondary term of size $x/(\log x)^{1+\delta +o(1)}$, where $\delta := 1 - \frac{1+\log \log 2}{\log 2} = 0.0860713320\dots $ is the multiplication table constant. Detailed heuristics suggest that this secondary term is asymptotic to $$ \frac{1}{\sqrt {\log\log x}} \cdot \frac x{(\log x)^{1+\delta }} $$ times a bounded, positive, $1$-periodic, non-constant function of $\frac{\log \log x}{\log 2}$.