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We significantly strengthen results on the structure of matrix rings overfinite 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 ifand 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 ofan odd prime and $F$ is a field of odd characteristic, having cardinalitystrictly 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 afinite field of cardinality $Q$.  A novel contribution to the ring-theoretical theme of this study is theclassification of finite fields $\FQ$ of odd order in which every element isthe sum of a tripotent and a potent. In this regard, we obtain an expressionfor the number of consecutive triples $\gamma-1,\gamma,\gamma+1$ of non-squareelements in $\FQ$; in particular, $\FQ$ contains three consecutive non-squareelements whenever $\FQ$ contains more than 9 elements.

Rings and finite fields whose elements are sums or differences of tripotents and potents