We show that every sufficiently large can be written as the sum of three primes, each of which is a sum of a square and a prime square. The main tools are a transference version of the circle method and various sieve related ideas. In particular, a majorant of the set of primes of interest is constructed that overestimates it by a factor of less than and for which we have good control of the Fourier transform.
Lasse Grimmelt. "Vinogradov’s theorem with Fouvry–Iwaniec primes." Algebra Number Theory 16 (7) 1705 - 1776, 2022. https://doi.org/10.2140/ant.2022.16.1705