  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.
