Summary: 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 \(\mathcal{A}\) 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 \(p(n_1) + m,\;p(n_2) + m \in \mathcal{A}\) for some \(m \in \mathcal{A}\).