It is possible to expand the Einstein field equations around their spherical symmetric solutions using so called tensor harmonics. Pioneering work on the topic has been done by T. Regge and J. A. Wheeler, Phys. Rev. 108, 1063 (1957), F. J. Zerilli, Phys. Rev. Lett. 24, 737 (1970) and many others.
In general the key idea is to expand the metric around the spherical symmetric one using spherical harmonics generalized to tensor and then solving the resulting field equations. In general this is not trivial and I do not know much about the general case.
I am however rather familiar with a special case or application of this approach: the so called Hartle-Thorne formalism: In a series of papers J. B. Hartle, K. S. Thorne and collaborators applied this approach to slowly rotating stars: expanding the metric in the angular velocity around the spherical symmetric one. This is a rather simple application and in its original form (up to second order in the angular velocity) it describes mono- and quadrupole perturbations away from spherical symmetry as well as an $l=1$ frame dragging term. The line element reads:
Treating $h_0$, $h_2$, $m_0$, $m_2$ and $k_2$ as small perturbations and $\omega$ as first order frame dragging term one can solve the Einstein field equations up to second order in the angular velocity. This approach can be used to describe not only deformations due to rotation but also to describe small deformations due to magnetic fields (eg. K. Ioka and M. Sasaki, Astrophys. J. 600:296-316 (2004)).
To conclude a few words on the specific question raised by OP: It is not a "simple" expansion but rather a complicated one and in general the geodesic equations can not be reduced back to a simple Hamiltonian with an effectiv potential in a $V(r)$ form.
There are analytic solutions for the metric perturbation (in case of the Hartle-Thone ansatz) but the geodesic equations become rather lengthy. They include effects like frame dragging as well as the perturbations away form spherical symmetry. The exterior solutions depend only on a few global parameters like, mass and radius of the unperturbed star and mass shift, angular momentum, angular velocity and mass quadrupole moment of the perturbed configuration. In that regard the geodesic equations are not that complicated in this case but that is just the case in this very special ansatz, which has very high symmetry and describes only the lowest order perturbation away from spherical symmetry and only for small perturbations.
For arbitrary large perturbations and higher "multipolarity" things become much more complicated and ultimately a more general treatment in the framework of Numerical Relativity (NR) becomes necessary.