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Some mean value theorems in the style of Bombieri-Vinogradov's theorem arediscussed. They concern binary and ternary additive problems with primes inarithmetic progressions and short intervals. Nontrivial estimates for some ofthese mean values are given. As application inter alia, we show that for largeodd n\not\equiv 1 (6), Goldbach's ternary problem n=p_1+p_2+p_3 is solvablewith primes p_1,p_2 in short intervals p_i \in [X_i,X_i+Y] withX_{i}^{\theta_i}=Y, i=1,2, and \theta_1,\theta_2\geq 0.933 such that(p_1+2)(p_2+2) has at most 9 prime factors.

Goldbach's problem with primes in arithmetic progressions and in short intervals