  There had been recent advancement toward the detection of ultra-faint dwarf
galaxies, which may serve as a useful laboratory for dark matter exploration
since some of them contains almost 99$\%$ of pure dark matter. The majority of
these galaxies contain no black hole that inhabits them. Recently, there had
been reports that some dwarf galaxies may have a black hole within. In this
study, we construct a black hole solution combined with the Dehnen dark matter
halo profile, which is commonly used for dwarf galaxies. We aim to find out
whether there would be deviations relative to the standard black hole
properties, which might allow determining whether the dark matter profile in an
ultra-faint dwarf galaxy is cored or cuspy. To make the model more realistic,
we applied the modified Newman-Janis prescription to obtain the rotating
metric. We analyzed the black hole properties such as the event horizon,
ergoregion, geodesics of time-like and null particles, and the black hole
shadow. Using these observables, the results indicate the difficulty of
distinguishing whether the dark matter is cored or cuspy. To find an observable
that can potentially distinguish these two profiles, we also calculated the
weak deflection angle to examine the effect of the Dehnen profile in finite
distance and far approximation. Our results indicate that using the weak
deflection angle is far better, in many orders of magnitude, in potentially
differentiating these profiles. We conclude that although dwarf galaxies are
dark matter-dominated places, the effect on the Dehnen profile is still
dependent on the mass of the black hole, considering the method used herein.
