Set $${T=N^{\frac{1}{3}-\epsilon}}$$ T = N 1 3 - ϵ . It is proved that for all but $${\ll TL^{-H},\,H > 0}$$ ≪ T L - H , H > 0 , exceptional prime numbers $${k\leq T}$$ k ≤ T and almost all integers b1, b2 co-prime to k, almost all integers $${n\sim N}$$ n ∼ N satisfying $${n\equiv b_{1}+b_{2}(mod\,k)}$$ n ≡ b 1 + b 2 ( m o d k ) can be written as the sum of two primes p1 and p2 satisfying $${p_{i}\equiv b_{i}(mod\,k),\,i=1,2}$$ p i ≡ b i ( m o d k ) , i = 1 , 2 . For prime numbers $${k\leq N^{\frac{5}{24}-\epsilon}}$$ k ≤ N 5 24 - ϵ , this result is even true for all but $${\ll (\log\,N)^{D}}$$ ≪ ( log N ) D primes k and all integers b1, b2 co-prime to k.