Representation of an integer as the sum of a prime in arithmetic progression and a square-free integer with parity on the number of prime factors

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 with an even (or odd) number of prime factors. Our method is based on the notion of local model developed by Ramar\'e and may be viewed as an abstract circle method.