Summary: We obtain the structure of the rings in which every element is either a sum or a difference of a nilpotent and an idempotent that commute. This extends the structure theorems of a commutative weakly nil-clean ring, of an abelian weakly nil-clean ring, and of a strongly nil-clean ring. As applications, this result is used to determine the 2-primal rings \(R\) such that the matrix ring \(\mathbb M_n(R)\) is weakly nil-clean, and to show that the endomorphism ring \(\mathrm{End}_D(V)\) over a vector space \(V_D\) is weakly nil-clean if and only if it is nil-clean or \(\dim(V)=1\) with \(D\cong\mathbb Z_3\).