Rings over which matrices are sums of idempotent and $q$-potent matrices

We study the rings over which each square matrix is the sum of an idempotent matrix and a $q$-potent matrix. We also show that if $F$ is a finite field not isomorphic to $\Bbb{F}_3$ and $q>1$ is odd then each square matrix over $F$ is the sum of an idempotent matrix and a $q$-potent matrix if and only if $q-1$ is divisible by $| F | -1$.