Suppose $M$ is a compact Riemannian manifold and let $d$ be the induced distance function on $M$. Let $\mu$ be a probability measure on $M$ with continuous density. The Fr$\acute{\mathrm{e}}$chet function is defined as
$$ F_{\mu}(x) = \int_{M} d^2(x,y) \,\mathrm{d}\mu(y) $$
I want to ask whether $F_{\mu}(x)$ is a smooth, or a $\mu$-almost surely smooth function on $M$? Does the regularity of $F_{\mu}(x)$ depend on the regularity of the density function of $\mu$?
For any fixed $x\in M$, the squared distance function $r_x(\cdot):=d^2(x,\cdot)$ is a $\mu$-almost surely smooth function (it is not differentiable on the cut locus of $x$). The gradient of $r_x$ is given by $\nabla r_x(y) = -2\exp_y^{-1}(x)$ (see this post). I feel that (might be wrong) the function $F_\mu(x)$ is continuously differentiable, and the gradient is given by
$$ \langle \nabla F(x),v \rangle_x = -2\int_M\langle \exp_x^{-1}(y),v \rangle_x\,\mathrm{d}\mu(y) $$
where $v\in T_xM$ is any tangent vector at $x$.
I'm also looking for examples where an explicity computation of $F_\mu(x)$ is possible, as a way to verify the above question. Thank you in advance!