Nearly Gorenstein cyclic quotient singularities

We investigate the nearly Gorenstein property among $d$-dimensional cyclic quotient singularities $\mathbb{k}[[x_1,\dots ,x_d]{]}^G$, where $\mathbb{k}$ is an algebraically closed field and $G\subseteq \mathrm{G}\mathrm{L}(d,\mathbb{k})$ is a finite small cyclic group whose order is invertible in $\mathbb{k}$. We prove a necessary and sufficient condition to be nearly Gorenstein that also allows us to find several new classes of such rings.