A bound for the nilpotence index associated to $m$-nil-clean nonderogatory matrices

It is proved that if $\mathbb{F}$ is a field of positive characteristic $p,$ and if $m$ and $n$ are positive integers such that $m\ge 2,$ and $n\le p\le mn-1,$ for every $n\times n$ nonderogatory matrix $A\in {\mathbb{M}}_n(\mathbb{F}),$ with trace in $\{k{.1}_{mathbbF}\mid k\in \{0,1,\dots ,p-1\}\},$ there exist $m$ idempotent matrices $E_1,E_2,\dots ,E_m,$ and a nilpotent matrix $N$, such that $A=E_1+E_2+\cdots +E_m+N,$ with $N^k=0,$ where $k=n$ if $p\in \{nm-1,nm-2\},$ $k=n-1$ if $n\in \{2,3\}$ or $p=nm-3,$ otherwise $k=\mathrm{m}\mathrm{a}\mathrm{x}(2,1+\lfloor \frac{n-1}{r}\rfloor ),$ if $n$ is even, and $k=\mathrm{m}\mathrm{a}\mathrm{x}(3,1+\lfloor \frac{n-1}{r}\rfloor ),$ if $n$ is odd, where $r:=\lfloor \frac{nm-p}{2}\rfloor \mathrm{.}$ Moreover, for $n>p,$ $A$ is the sum of two idempotent matrices, and a square zero one, if $n$ is even, and it is sum of two idempotent matrices and one which third power is zero, if $n$ is odd.