On the sum of a prime and a square-free number co-prime to any integer with at most two prime factors

Every natural number greater than $2$ can be written as the sum of a prime and a square-free number, and recent work has imposed additional divisibility conditions on the square-free number. We overcome limitations in these works to prove new results on square-free numbers co-prime to any integer with up to two prime factors, which make the expected asymptotic results explicit.