Non-rational divisors over non-degenerate $cDV$-points

Suppose that $(X,o)$ is a 3-dimensional terminal singularity of type $cD$ or $cE$ defined in ${\mathbb C}^4$ by an equation that is non-degenerate with respect to its Newton diagram. We show that there exists at most one non-rational divisor $E$ over $(X,o)$ with discrepancy $a(E,X)=1$. We also describe all the blow-ups of the singularity $(X,o)$ with non-rational exceptional divisors of discrepancy 1.