Intersections of binary quadratic forms in primes and the paucity phenomenon

Let r(n) be the function that counts the number of ways to represent a natural number n as a sum of two positive squares. I am going to talk about my work in progress about second moments for r(n), when some of the variables are restricted to primes and discuss the existence of paucity for the off-diagonal solutions in such problems. The methods are largely based on Hooley technique to tackle (on GRH) the Hardy-Littlewood problem about the representations N=p+a^2+b^2, where p is a prime and a, b are integers, some related works of Plaksin (based on the unconditional resolution of the Hardy-Littlewood problem by Linnik) and more recent results of Daniel.