  Let $K$ be a field of characteristic $p$, $\delta$ a nonzero
$\mathcal{E}$-derivation and $D=f(x_1)\partial_1$. We first prove that
$\operatorname{Im}D$ is not a Mathieu-Zhao space of $K[x_1]$ if and only if
$f(x_1)=x_1^rf_1(x_1^p)$ and $r\neq 1$. Then we prove that
$\operatorname{Im}\delta$ is a Mathieu-Zhao space of $K[x_1]$ if and only if
$\delta$ is not locally nilpotent. Finally, we classify some nilpotent
derivations of $K[x_1]$ and give a sufficient and necessary condition for
$D(I)$ to be a Mathieu-Zhao space of $K[x_1]$ for any ideal $I$ of $K[x_1]$.
