  For $n\ge 2$ and fixed $k\ge 1$, we study when a square matrix $A$ over an
arbitrary field $\mathbb{F}$ can be decomposed as $T+N$ where $T$ is a torsion
matrix and $N$ is a nilpotent matrix with $N^k=0$. For fields of prime
characteristic, we show that this decomposition holds as soon as the
characteristic polynomial of $A\in \mathbb{M}_{n}(\mathbb{F})$ is algebraic
over its base field and the rank of $A$ is at least $\frac nk$, and we present
several examples that show that the decomposition does not hold in general.
Furthermore, we completely solve this decomposition problem for $k=2$ and
nilpotent matrices over arbitrary fields (even over division rings). This
somewhat continues our recent publications in Lin. \& Multilin. Algebra (2023)
and Internat. J. Algebra \& Computat. (2022) as well as it strengthens results
due to Calugareanu-Lam in J. Algebra \& Appl. (2016).
