  Let $G$ be a 2-connected $n$-vertex graph and $N_s(G)$ be the total number of
$s$-cliques in $G$. Let $k\ge 4$ and $s\ge 2$ be integers. In this paper, we
show that if $G$ has an edge $e$ which is not on any cycle of length at least
$k$, then $N_s(G)\le r{k-1\choose s}+{t+2\choose s}$, where $n-2=r(k-3)+t$ and
$0\le t\le k-4$. This result settles a conjecture of Ma and Yuan and provides a
clique version of a theorem of Fan, Wang and Lv. As a direct corollary, if
$N_s(G)> r{k-1\choose s}+{t+2\choose s}$, every edge of $G$ is covered by a
cycle of length at least $k$.
