  The detection of gravitational waves from Extreme mass Ratio Inspirals
(EMRIs) by the future space-based gravitational-wave detectors demands the
generation of accurate enough waveform templates. Since the spin of the smaller
secondary body cannot be neglected for the detection and parameter estimation
of EMRIs, we study its influence on the phase of the gravitational waves from
EMRIs with spinning secondary. We focus on generic eccentric equatorial orbits
around a Kerr black hole. To model the spinning secondary object, we use the
Mathisson-Papapetrou-Dixon equations in the pole-dipole approximation.
Furthermore, we linearize in spin the orbital variables and the
gravitational-wave fluxes from the respective orbits. We obtain these fluxes by
using the Teukolsky formalism in the frequency domain. We derive the evolution
equations for the spin induced corrections to the adiabatic evolution of an
inspiral. Finally, through their numerical integration we find the
gravitational-wave phase shift between an inspiral of a spinning and a
non-spinnig body.
