  In this paper, we compute the size of the exceptional set in a generalized
Goldbach problem and show that for a given polynomial $f(x) \in \mathbb{Z}[x]$
with a positive leading coefficient, positive integers $A$, $B$, $g$ and $0
\leq i, j < g$, there are infinitely many integers $n$ which satisfy $f(n) = Ap
+ Bq$ for some primes $p \equiv i, q \equiv j \pmod{g}$ under a mild condition.
