Non-rational divisors over non-degenerate cDV-points

Let $(X,o)$ be a 3-dimensional terminal singularity of type $cD$ or $cE$ defined in ${\mathbb{C}}^4$ by an equation non-degenerate with respect to its Newton diagram. We show that there is not more than 1 non-rational divisor $E$ over $(X,o)$ with discrepancy $a(E,X)=1$. We also describe all blowups $\sigma$ of $(X,o)$ such that $E=\text{Exc}(\sigma )$ is non-rational and $a(E,X)=1$.