  We study extreme mass ratio inspirals occurring in modified gravity, for
which the system is modeled by a small compact object with scalar charge
spiraling into a supermassive Kerr black hole. Besides the tensorial
gravitational waves arising from the metric perturbations, radiations are also
induced by the scalar field. The relevant metric and scalar perturbations are
triggered by the orbital motion of the small object, which give rise to a
system of inhomogeneous differential equations under the adiabatic
approximation. Such a system of equations is then solved numerically using
Green's function furnished by the solutions of the corresponding homogeneous
equations. To explore the present scenario from an observational perspective,
we investigate how the pertinent observables are dependent on specific
spacetime configurations. In this regard, the energy fluxes and the
gravitational wave dephasing accumulated during the process are evaluated, as
functions of the scalar charge, mass ratio, and spin of the central
supermassive black hole. In particular, the presence of additional scalar
emission leads to a more significant rate of overall energy loss, which, in
turn, decreases the total number of orbital cycles before the small object
plunges into the central black hole. Moreover, for a central black hole with a
higher spin, the imprints of the scalar charge on the resultant gravitational
radiations are found to be more significant, which indicates the possibility of
detecting the scalar charge.
