  In 2018, Kalck and Yang showed that the singularity categories associated
with 3-dimensional Gorenstein quotient singularities are triangle equivalent
(up to direct summands) to small cluster categories associated with McKay
quivers with potential. We introduce graded McKay quivers with potential and
generalize Kalck-Yang's theorem to arbitrary dimensions. The singularity
categories we consider occur as stable categories of categories of maximal
Cohen-Macaulay modules. We refine our description of the singularity categories
by showing that these categories of maximal Cohen-Macaulay modules are
equivalent to Higgs categories in the sense of Wu. Moreover, we describe the
singularity categories in the non-Gorenstein case.
