Why K-harmonic means is less sensitive to random initialization than K-means?
Harmonic mean is less sensitive to outliers.

Say that you have 3 numbers: 1 1 and 30001

Their arithmetic mean (average) is 10001 and their harmonic mean is 2 that is much closer to the majority of the points.

So in a way, using an harmonic mean is more forgiving.

A Bayesian look at the harmonic mean

Assuming that the closer a point is, the more likely it is to be in a cluster.

One can choose to model the distribution of distances from an origin point, as an Gamma-Exponential model with a priors $\alpha=m$ and $\beta=\frac{m}{\mu_0}$ where $\mu_0$ is the initial mean and $m$ is the strength of the prior.

And given observations of $\{d_1,d_2,\dots,d_n\}$, the updated model's mean is:
$$\mu'=\frac{m+n}{\frac{m}{\mu_0}+\frac{1}{d_1}+\frac{1}{d_2}+\dots+\frac{1}{d_n}}$$
We got our beloved harmonic mean.

You could argue that the harmonic mean in k-means context is an assumption on the distribution of distances.