By using the closure concept introduced by the last author, we prove that for any sufficiently large nonhamiltonian claw-free graph G satisfying a degree condition of type σ k ( G)> n+ k 2−4 k+7 (where k is a constant), the closure of G can be covered by at most k−1 cliques. Using structural properties of the closure concept, we show a method for characterizing all such nonhamiltonian exceptional graphs with limited clique covering number. The method is explicitly carried out for k⩽6 and illustrated by proving that every 2-connected claw-free graph G of order n⩾77 with δ( G)⩾14 and σ 6( G)> n+19 is either hamiltonian or belongs to a family of easily described exceptions.