One of the fundamental differences between the Central Limit Theorem for empirical Fréchet means obtained in [Kendall and Le, 'Limit theorems for empirical Fréchet means of independent and non-identically distributed manifold-valued random variables ', Braz. J. Probab. Stat. 25 (2011) 323-352] and that for empirical Euclidean means lies on the assumption that the probability measure of the cut locus of the true Fréchet mean is zero. In [Hotz and Huckemann, 'Intrinsic means on the circle: uniqueness, locus and asymptotics', Preprint, 2011, arXiv:1108.2141v1], the authors show that, in the case of a circle, this assumption holds automatically. This paper shows that this holds for any complete and connected Riemannian manifold, assuming that there are at least two minimal geodesics between the Fréchet mean and any point in its cut locus and that, in fact, it can also be generalized to local minima of the p-energy function of a finite measure.