Sufficient condition for $xy^*z \subseteq L$ for a DFA with $n$ states
Let the automaton have states $Q$, initial state $q_0$, accepting states $F$, and transition function $\delta$, and suppose that $|Q| = n$. Denote the language that it accepts by $L$.
Suppose that $xy^iz \in L$ for $i = 0,\ldots,n-1$. Denote $p_i = \delta(q_0,xy^i)$. We consider two cases:

The states $p_0,\ldots,p_{n-1}$ are all different. In this case, $Q = \{p_0,\ldots,p_{n-1}\}$, and so $\delta(q,z) \in F$ for all $q \in Q$, implying that $\delta(q_0,xy^iz) \in F$ for all $i$, and so $xy^iz \in L$ for all $i$.
For some $j &lt; k$, we have $p_j = p_k$. It follows that the sequence $p_i$ is eventually periodic, and so again $xy^iz \in L$ for all $i$.