Given a clique-width expression of a graph $G$ of clique-width $k$, we provide $2^{O(k)}\cdot n^{O(1)}$ time algorithms for connectivity constraints on locally checkable properties such as Connected Dominating Set, Connected Perfect Dominating Set or Node-Weighted Steiner Tree. We also propose an $2^{O(k)}\cdot n^{O(1)}$ time algorithm for Feedback Vertex Set. The best running times for all the considered cases were either $2^{O(k\cdot \log(k))}\cdot n^{O(1)}$.