  Circumambient and galactic-scale environments are intermittently present
around black holes that reside in active galactic nuclei. As supermassive black
holes impart energy on their host galaxy, so the galactic environment affects
the dynamics of solar-mass objects around black holes and the gravitational
waves emitted from non-vacuum asymmetric binaries. Only recently an exact
general-relativistic solution has been found that describes a Schwarzschild
black hole immersed in a dark matter halo of the Hernquist type. We perform an
extensive analysis of generic geodesics delving in such non-vacuum spacetimes
and compare our results with those obtained in Schwarzschild, as well as
calculate their gravitational-wave emission. Our findings indicate that the
radial and polar oscillation frequency ratios descend deeper into the strong
gravity region as the compactness of the halo increases. This translates to a
redshift of non-vacuum geodesics and their resulting waveforms with respect to
the vacuum ones. We calculate the overlap between waveforms resulting from
Schwarzschild and non-vacuum geometries and find that it decreases as the halo
compactness grows, meaning that dark matter environments should be
distinguishable by space-borne detectors. For compact environments, we find
that the apsidal precession is strongly affected due to the gravitational pull
of dark matter; the orbit's axis can rotate in the opposite direction as that
of the orbital motion, leading to a retrograde precession drift that depends on
the halo mass, as opposed to the typical prograde precession transpiring in
galactic-scale environments. Gravitational waves in retrograde-to-prograde
alterations demonstrate transient frequency phenomena around critical
non-precessing turning points, thus they may serve as `smoking guns' for the
presence of compact dark matter environments around supermassive black holes.
