  The discovery of gravitational waves and black holes has started a new era of
gravitational wave astronomy that allows us to probe the underpinning features
of gravity and astrophysics in extreme environments of the universe. In this
article, we investigate one such study with an extreme mass-ratio inspiral
system where the primary object is a spherically symmetric static black hole
immersed in a dark matter halo governed by the Hernquist density distribution.
We consider the eccentric equatorial orbital motion of the steller-mass object
orbiting around the primary and compute measurable effects. We examine the
behaviour of dark matter mass and halo radius in generated gravitational wave
fluxes and the evolution of eccentric orbital parameters -- eccentricity and
semi-latus rectum. We further provide an estimate of gravitational wave
dephasing and find the seminal role of low-frequency detectors in the
observational prospects of such an astrophysical environment.
