For a graph G, let σ̄k+3(G)=min{d(x1)+d(x2)+⋯+d(xk+3)−|N(x1)∩N(x2)∩⋯∩N(xk+3)|∣x1,x2,…,xk+3 are k+3 independent vertices in G}. In [H. Li, On cycles in 3-connected graphs, Graphs Combin. 16 (2000) 319–335], H. Li proved that if G is a 3-connected graph of order n and σ̄4(G)≥n+3, then G has a maximum cycle such that each component of G−C has at most one vertex. In this paper, we extend this result as follows. Let G be a (k+2)-connected graph of order n. If σ̄k+3(G)≥n+k(k+2), G has a cycle C such that each component of G−C has at most k vertices. Moreover, the lower bound is sharp.