It is proved that for almost all prime numbers $k\le N^{1/4-ϵ}$, any fixed integer b₂, (b₂,k) = 1, and almost all integers b₁, 1 ≤ b₁ ≤ k, (b₁,k) = 1, almost all integers n satisfying n ≡ b₁ + b₂ (mod k) can be written as the sum of two primes p₁ and p₂ satisfying $p_i\equiv b_i\left(modk\right)$, i = 1,2. For the proof of this result, new estimates for exponential sums over primes in arithmetic progressions are derived.