Let $M$ be a spherically symmetric $C^2$ manifold. Consider the open ball $B_{R}$, centered in $x\in M$. Let $y\in B_{R}$, then we define by $\rho$ the geodesic distance, beetwen $x$ and $y$.
I don't see the term $-\frac{1}{2}h_{p}$. Sorry if this is very imprecise, but my point is that I do not understand the indexes of the derivatives of $(*)$, but for me it is not correct that in the derivative the same argument of the function appears and even more multiplying (for instance $h_{p}(p,\theta)p$). Thanks!!!!
A full 3-dimensional power series, centered at $x$... that's not something you want to put into spherical coordinates (centered at $x$). The angle variables are singular there. The radius increases in every direction from there. The fundamental premise of the series requires an independent system of coordinates, and you can't have that when the series is centered at the polar origin.
So, what are we really doing here? We're treating this as a single-variable problem. That $h_{\rho}$ isn't the derivative with respect to radius, it's a directional derivative in the (fixed) $\rho$ direction. The series is based entirely on the values of the function along that line, and is used only to calculate values on that line. It is not a Taylor series for the full three-dimensional function.