This paper studies the relationship between the degree of subgraphs and Hamiltonicity of graphs. It is proven that every 3-connected claw-free graph $G$ of order $n$ with minimum degree $\delta(G)\geq4$ is Hamilton-connected if it satisfies $d(H_1)+d(H_2)\geq n$ for any two non-adjacent subgraphs $H_1$, $H_2$ which are isomorphic to $P_4$, $K_1$ respectively.