Summary: Given a symmetric variety \(Y\) defined over \(\mathbb{Q}\) and a non-zero polynomial with integer coefficients, we use techniques from homogeneous dynamics to establish conditions under which the polynomial can be made \(r\)-free for a Zariski dense set of integral points on \(Y\). We also establish an asymptotic counting formula for this set. In the special case that \(Y\) is a quadric hypersurface, we give explicit bounds on the size of \(r\) by combining the argument with a uniform upper bound for the density of integral points on general affine quadrics defined over \(\mathbb{Q}\).