Non-rational divisors over non-Gorenstein terminal singularities

Let $(X,o)$ be a germ of a 3-dimensional terminal singularity of index $m\geq 2$. If $(X,o)$ has type cAx/4, cD/3-3, cD/2-2, or cE/2, then assume that the standard equation of $X$ in $\mathbb{C}^4/\mathbb{Z}_m$ is non-degenerate with respect to its Newton diagram. Let $π\colon Y\to X$ be a resolution. We show that there are not more than 2 non-rational divisors $E_i$, $i=1,2$, on $Y$ such that $π(E_i)=o$ and discrepancy $a(E_i,X)\leq 1$. When such divisors exist, we describe them as exceptional divisors of certain blowups of $X$ and study their birational type.