In this work we introduce and analyze a novel pressure-robust Hybrid High- Order method for the steady incompressible Navier–Stokes equations on general meshes. The proposed method supports arbitrary approximation orders, and is (relatively) inexpensive thanks to the possibility of statically condensing a subset of the unknowns at each nonlinear iteration. For regular solutions and under a standard data smallness assumption, we prove a pressure-independent energy error estimate on the velocity of order (k + 1). More precisely, when polynomials of degree k ≥ 0 at mesh elements and faces are used, this quantity is proved to converge as h^(k+1) (with h denoting the meshsize). The proposed method is a direct extension of a previous work done by the same authors using simplicial meshes. Numerical results are presented to support the theoretical analysis.