A graph polynomial $P$ is weakly distinguishing if for almost all finite graphs $G$ there is a finite graph $H$ that is not isomorphic to $G$ with $P(G)=P(H)$. It is weakly distinguishing on a graph property $\mathcal{C}$ if for almost all finite graphs $G\in\mathcal{C}$ there is $H \in \mathcal{C}$ that is not isomorphic to $G$ with $P(G)=P(H)$. We give sufficient conditions on a graph property $\mathcal{C}$ for the characteristic, clique, independence, matching, and domination and $ξ$ polynomials, as well as the Tutte polynomial and its specialisations, to be weakly distinguishing on $\mathcal{C}$. One such condition is to be addable and small in the sense of C. McDiarmid, A. Steger and D. Welsh (2005). Another one is to be of genus at most $k$.