Let $r(z)$ denotes the number of solutions in positive integers to $x+y\leq z$ with the unknown $x,y$ belonging to a set $S$ satisfying the following: the number of elements in $S$ less or equal to $x$ is asymptotically

$$N_S(x) \sim \frac{a x^b}{(\log x)^c}, \mbox{ with } 0<a, 0<b<1, \mbox{ and } c>0.$$

Then we have:

$$r(z) \sim \frac{a^2b z^{2b}}{(\log z)^{2c}}\cdot \int_0^1 (1-v)^b v^{b-1}dv = \frac{a^2b z^{2b}}{(\log z)^{2c}}\cdot \frac{\Gamma(b)\Gamma(b+1)}{\Gamma(2b+1)}$$

This covers sums of two squares and sums of two primes. It has implications about the possibility to solve Goldbach's conjecture, see the third answer in my previous MathOverflow question, here.