  Direct detection of gravitational waves and binary black hole mergers have
proven to be remarkable investigations of general relativity. In order to have
a definitive answer as to whether the black hole spacetime under test is the
Kerr or non-Kerr, one requires accurate mapping of the metric. Since EMRIs are
perfect candidates for space-based detectors, Laser Interferometer Space
Antenna (LISA) observations will serve a crucial purpose in mapping the
spacetime metric. In this article, we consider such a study with the Johannsen
spacetime that captures the deviations from the Kerr black hole and further
discuss their detection prospects. We analytically derive the leading order
post-Newtonian corrections in the average loss of energy and angular momentum
fluxes generated by a stellar-mass object exhibiting eccentric equatorial
motion in the Johannsen background. We further obtain the orbital evolution of
the inspiralling object within the adiabatic approximation and estimate the
orbital phase. We lastly provide the possible detectability of deviations from
the Kerr black hole by estimating gravitational wave dephasing and highlight
the crucial role of LISA observations.
