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We investigate polynomial patterns which can be guaranteed to appear in  weakly mixing  sets introduced by Furstenberg and studied by Fish. In particular, we prove that if   $\mathscr {A}$   is a weakly mixing set and   $p(x) \in \mathbb {Z}[x]$   is a polynomial of odd degree with positive leading coefficient, then all sufficiently large integers   $N$   can be represented as   $N = n_1 + n_2$  , where   $p(n_1) + m,\ p(n_2) + m \in \mathscr {A}$   for some   $m \in \mathscr {A}$  .

Weakly mixing sets of integers and polynomial equations