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 C4 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.