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