Fields over which matrices can be represented as the sum of potent and nilpotent matrices

We study fields over which every matrix can be represented as a sum of two potent matrices and a nilpotent matrix. In particular, it is shown that over a field $P$ every matrix can be represented as a sum of two idempotent matrices and a nilpotent matrix exactly when either $P\cong \mathbb{F}_2,$ or $P\cong \mathbb{F}_3$.