This paper is devoted to investigating the fundamental properties of high-order proximal operator (HOPE) and high-order Moreau envelope (HOME) in the nonconvex setting, meaning that the quadratic regularization (p=2) is replaced with a regularization with p>1. After studying several basic properties of HOPE and HOME, we investigate the differentiability and weak smoothness of HOME under q-prox-regularity q≥2 and p-calmness for p∈(1,2] and 2≤p≤q. Further, we design of a high-order proximal-point algorithm (HiPPA) for which the convergence of the generated sequence to proximal fixed points is studied. Our results pave the way toward the high-order smoothing theory with p>1 that can lead to algorithmic developments in the nonconvex setting, where our numerical experiments of HiPPA on Nesterov-Chebyshev-Rosenbrock functions show the potential of this development for nonsmooth and nonconvex optimization.