  We investigate the ground-state phase diagram of a modified spinless
Haldane-Hubbard model with broken threefold rotational symmetry, employing
exact diagonalization calculations. The interplay of asymmetry, interactions,
and topology gives rise to a rich phase diagram. The non-interacting limit of
the Hamiltonian exhibits a higher-order topological insulator characterized by
the existence of corner modes, in contrast to known chiral edge metallic states
of the standard Haldane model. Our investigation demonstrates that these
symmetry-protected states are robust to the presence of finite interactions.
Furthermore, in certain regimes of parameters, we show that a topological Mott
insulator exists in this model, where a non-trivial topological bulk coexists
with an interaction-driven charge-density-wave, whose emergence is
characterized by a $Z_2$-symmetry breaking within the 3$d$-Ising universality
class.
