$m$-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 $m\geq2,$ and $n\leq p\leq mn-1,$ for every $n\times n$ nonderogatory matrix $A\in \mathbb{M}_n(\mathbb{F}),$ with trace in $\{k.1_{\mathbb{F}}\mid k\in \{0,1,\dots,p-1\}\},$ there exist $m$ idempotent matrices $E_1, E_2,\dots, E_m,$ and a nilpotent matrix $N$, such that $A=E_1+E_2+\dots+E_m+N,$ with $N^k=0,$ where $k=n$ if $p\in \{nm-1,nm-2\},$ $k=n-1$ if $p=nm-3,$ otherwise $k=\mathrm{max}(2,1+\lfloor\frac{n-1}{r}\rfloor),$ if $n$ is even, and $k=\mathrm{max}(3,1+\lfloor\frac{n-1}{r}\rfloor),$ if $n$ is odd, where $r:=\lfloor\frac{nm-p}{2}\rfloor.$ 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.