A mean value of the representation function for the sum of two primes in arithmetic progressions

In this note, assuming a variant of the Generalized Riemann Hypothesis, which does not exclude the existence of real zeros, we prove an asymptotic formula for the mean value of the representation function for the sum of two primes in arithmetic progressions. This is an improvement of the result of F. R"uppel in 2009, and the generalization of the result of A. Languasco and A. Zaccagnini concerning the ordinary Goldbach problem in 2012.