Erdős problem 351

Let $p(x) \in \mathbb{Q}[x]$ be a non-constant rational polynomial with positive leading coefficient. Is it true that $$A=\{ p(n)+1/n : n \in \mathbb{N}\}$$ is strongly complete, in the sense that, for any finite set $B$, $$\left\{\sum_{a \in X} a : X \subseteq A \setminus B, X \textrm{ is finite}\right\}$$ contains all sufficiently large integers?