  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.
