Uniformly for small $q$ and $(a,q)=1$, we obtain an estimate for the weighted number of ways a sufficiently large integer can be represented as the sum of a prime congruent to $a$ modulo $q$ and a square-free integer. Our method is based on the notion of local model developed by Ramaré and may be viewed as an abstract circle method.
"Representation of an integer as the sum of a prime in arithmetic progression and a square-free integer." Funct. Approx. Comment. Math. 64 (1) 77 - 108, March 2021. https://doi.org/10.7169/facm/1896