  In the scope of black hole spectroscopy, several attempts have been made in
the past decades in order to test black holes or gravitational theories via
black hole quasinormal modes. In the eikonal approximation, the quasinormal
modes are generically associated with the photon ring of the black hole. This
correspondence is valid for most cases in general relativity, but may not be
true in other theories of gravity. In this paper, we consider the generalized
energy-momentum squared gravity in which matter fields are non-minimally
coupled to geometry. We investigate the axial perturbations of the charged
black holes in this model, without assuming any explicit expression of the
action functional. After obtaining the modified Klein-Gordon equation and the
modified Maxwell equations, we perturb the gravitational equations and the
modified Maxwell equations to derive the coupled master equations of the axial
perturbations. In the presence of the non-minimal coupling between matter and
geometry, the correspondence between the eikonal quasinormal modes and the
photon ring is not satisfied in general. Also, the two coupled fields of the
axial perturbations are found to propagate independently and they do not share
the same quasinormal frequencies in the eikonal limit.
