$m$-nil-clean nonderogatory matrices

We prove that if $\mathbb{F}$ is a field of positive odd characteristic $p,$ and $m,$ and $n$ are positive integers such that $m\ge 2,$ and $n\le p,$ every $n\times n$ nonderogatory matrix $A\in {\mathbb{M}}_n(\mathbb{F})$ which is sum of $m$ idempotents and a nilpotent, has a decomposition $A=E_1+E_2+\cdots +E_m+V,$ such that $E_i^2=E_i,$ for every $i\in \{1,\dots ,m\},$ and $V^{[\frac{p-2}{m}]+2}=0.$