  The geometric optics approximation provides an interpretation for eikonal
correspondence that, in black-hole-containing spacetimes, connects
high-frequency black hole quasinormal modes with closed photon orbits around
said black hole. This correspondence has been identified explicitly for
Schwarzschild, Reissner-Nordstr\"om, Kerr, and Kerr-Newman black holes, the
violation of which can be a potential hint toward physics beyond General
Relativity. Notably, the aforementioned black hole spacetimes have sufficient
symmetries such that both the geodesic equations and the master wave equations
are separable. The identification of the correspondence seems to largely rely
on these symmetries. One naturally asks how the eikonal correspondence would
appear if the spacetime were less symmetric. For a pioneering work in this
direction, we consider in this paper a deformed Schwarzschild spacetime
retaining only axisymmetry and stationarity. We show that up to the first order
of spacetime deformations the eikonal correspondence manifests through the
definition of the \textit{averaged} radius of trapped photon orbits along their
one period. This averaged radius overlaps the potential peak in the master wave
equation, which can be defined up to the first order of spacetime deformations,
allowing the explicit identification of the eikonal correspondence.
