  Some mean value theorems in the style of Bombieri-Vinogradov's theorem are
discussed. They concern binary and ternary additive problems with primes in
arithmetic progressions and short intervals. Nontrivial estimates for some of
these mean values are given. As application inter alia, we show that for large
odd n\not\equiv 1 (6), Goldbach's ternary problem n=p_1+p_2+p_3 is solvable
with primes p_1,p_2 in short intervals p_i \in [X_i,X_i+Y] with
X_{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.
