Let $k$ be a positive integer. A graph is said to be $(P_2 \cup kP_1)$-free if it does not contain $P_2 \cup kP_1$ as an induced subgraph. Recently, Ota and the author asked whether every 1-tough and $k$-connected $(P_2 \cup kP_1)$-free graph is hamiltonian or the Petersen graph. Note that this problem is affirmative for $k \in \{1,2,3\}$ by the known results. In this paper, we show that for each integer $k \geq 4$, if $G$ is a $1$-tough and $(k-1)$-connected $(P_2 \cup kP_1)$-free graph with $|V(G)| \ge k^2+k+1$ and $δ(G) \ge k$, then $G$ is hamiltonian. This result implies that the above question is affirmative for large graphs.