The exceptional set in a generalized Goldbach`s problem

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.

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