Summary: We introduce a rectified \(n\)-harmonic map flow from an \(n\)-dimensional closed Riemannian manifold to another closed Riemannian manifold. We prove existence of a global solution, which is regular except for a finite number of points, of the rectified \(n\)-harmonic map flow and establish an energy identity for the flow at each singular time. Finally, we present two applications of the rectified \(n\)-harmonic map flow to minimizing the \(n\)-energy functional and the Dirichlet energy functional in a homotopy class.