  We examine extreme mass ratio inspirals (EMRIs), where a charged compact
object spirals into a supermassive black hole, in modified gravity theories
with additional scalar or vector fields. Using the Teukolsky and generalized
Sasaki-Nakamura formalisms, we provide the post-Newtonian expansion of the
energy flux of the vector waves up to $O(v^5)$ beyond the quadrupole formula in
the weak field and numerically calculate the energy flux in the strong field
for a charged particle moving in circular orbits. Our findings reveal a
degeneracy in the scalar and vector charge parameters for weak-field,
slow-motion orbits. However, for strong-field, fast-motion orbits close to the
innermost stable circular orbit, we observe distinct behaviors between scalar
and vector fields. We investigate the potential of using EMRIs detected by
space-based gravitational-wave detectors, such as the Laser Interferometer
Space Antenna to identify whether a black hole carries a scalar or vector
charge. The influence of scalar and vector flux on the orbital evolution and
tensor GW phase can not help us distinguish scalar and vector fields. However,
extra polarizations emitted by the scalar or vector field can break the
correlations between the scalar field and vector field and then help us
distinguish the scalar and vector field.
