Nil-clean and strongly nil-clean rings by Tamer Koşan, Zhou Wang, and Yiqiang Zhou.
Theorem 2.1. An element $a\in R$ is strongly nil-clean iff $a$ is strongly clean in $R$ and $a−a^2$ is a nilpotent.
Proof (⇒) . Let $a=e+b$ be a strongly nil-clean decomposition in $R$. Then $a=(1−e)+(2e−1+b)$ is a strongly clean decomposition in $R$.
Why does $2e-1+b$ is a unit?
Definitions
$a$ is said to be strongly nil-clean if $a=e+b$ where $e$ is idempotent and $b$ is nilpotent, and $eb=be$.
$a$ is said to be strongly clean if $a=e+b$ where $e$ is idempotent, $b$ is a unit, and $eb=be$.
By hypothesis, $a=e+b$ with $e$ idempotent and $b$ nilpotent, and $eb=be$.
$2e-1$ is a unit with inverse $2e-1$, and it commutes with the nilpotent element $b$. By a well-known lemma, their sum is a unit.