  We significantly strengthen results on the structure of matrix rings over
finite fields and apply them to describe the structure of the so-called weakly
$n$-torsion clean rings. Specifically, we establish that, for any field $F$
with either exactly seven or strictly more than nine elements, each matrix over
$F$ is presentable as a sum of of a tripotent matrix and a $q$-potent matrix if
and only if each element in $F$ is presentable as a sum of a tripotent and a
$q$-potent, whenever $q>1$ is an odd integer. In addition, if $Q$ is a power of
an odd prime and $F$ is a field of odd characteristic, having cardinality
strictly greater than $9$, then, for all $n\geq 1$, the matrix ring
$\mathbb{M}_n(F)$ is weakly $(Q-1)$-torsion clean if and only if $F$ is a
finite field of cardinality $Q$.
  A novel contribution to the ring-theoretical theme of this study is the
classification of finite fields $\FQ$ of odd order in which every element is
the sum of a tripotent and a potent. In this regard, we obtain an expression
for the number of consecutive triples $\gamma-1,\gamma,\gamma+1$ of non-square
elements in $\FQ$; in particular, $\FQ$ contains three consecutive non-square
elements whenever $\FQ$ contains more than 9 elements.
