  In the $K_r$-Cover problem, given a graph $G$ and an integer $k$ one has to
decide if there exists a set of at most $k$ vertices whose removal destroys all
$r$-cliques of $G$. In this paper we give an algorithm for $K_r$-Cover that
runs in subexponential FPT time on graph classes satisfying two simple
conditions related to cliques and treewidth. As an application we show that our
algorithm solves $K_r$-Cover in time
  * $2^{O_r\left (k^{(r+1)/(r+2)}\log k \right)} \cdot n^{O_r(1)}$ in
pseudo-disk graphs and map-graphs;
  * $2^{O_{t,r}(k^{2/3}\log k)} \cdot n^{O_r(1)}$ in $K_{t,t}$-subgraph-free
string graphs; and
  * $2^{O_{H,r}(k^{2/3}\log k)} \cdot n^{O_r(1)}$ in $H$-minor-free graphs.
