We consider the difference operator HW = U + U−1 + W, where U is the self-adjoint Weyl operator U = e−bP, b > 0, and the potential W is of the form W(x) = x2N + r(x) with N∈ℕ and |r(x)| ≤ C(1 + |x|2N−ɛ) for some 0 < ɛ ≤ 2N − 1. This class of potentials W includes polynomials of even degree with leading coefficient 1, which have recently been considered in Grassi and Mariño [SIGMA Symmetry Integrability Geom. Methods Appl. 15, 025 (2019)]. In this paper, we show that such operators have discrete spectrum and obtain Weyl-type asymptotics for the Riesz means and for the number of eigenvalues. This is an extension of the result previously obtained in Laptev et al. [Geom. Funct. Anal. 26, 288–305 (2016)] for W = V + ζV−1, where V = e2πbx, ζ > 0