  We show that for every fixed $A>0$ and $\theta>0$ there is a
  $\vartheta=\vartheta(A,\theta)>0$ with the following property.
  Let $n$ be odd and sufficiently large, and let
  $Q_{1}=Q_{2}:=n^{\h}(\log n)^{-\vartheta}$ and $Q_{3}:=(\log n)^{\theta}$.
Then for all $q_{3}\leq Q_{3}$, all reduced residues $a_{3}$ mod $q_{3}$,
almost all $q_{2}\leq Q_{2}$, all admissible residues $a_{2}$ mod $q_{2}$,
almost all $q_{1}\leq Q_{1}$ and all admissible residues $a_{1}$ mod $q_{1}$,
there exists a representation $n=p_{1}+p_{2}+p_{3}$ with primes $p_{i}\equiv
a_{i} (q_{i})$, $i=1,2,3$.
