We derive novel error estimates for Hybrid High-Order (HHO) discretizations of Leray-Lions problems set in $W^{1, p}$ with $p\in(1, 2]$. Specifically, we prove that, depending on the degeneracy of the problem, the convergence rate may vary between $(k + 1)(p − 1)$ and $(k + 1)$, with $k$ denoting the degree of the HHO approximation. These regime-dependent error estimates are illustrated by a complete panel of numerical experiments.