  We explore topological properties of a modulated Haldane model (MHM) in which
the strength of the nearest-neighbor and next-nearest-neighbor terms is made
unequal and the three-fold rotational symmetry $\mathcal{C}_3$ is broken by
introducing a trimerization term ($|t_{1w(2w)}|< t_{1s(2s)}$) in the
Hamiltonian. Using the parameter $\eta={t_{1w}}/{t_{1s}}= t_{2w}/t_{2s}$, we
show that the MHM supports a transition from the quantum anomalous Hall
insulator (QAHI) to a HOTI phase at $\eta=\pm 0.5$. The MHM also hosts a
zero-energy corner mode on a nano-disk that can transition to a trivial
insulator without gap-closing when the inversion symmetry is broken. The
gap-closing critical states are found to be magnetic semimetals with a single
Dirac node which, unlike the classic Haldane model, can move along the
high-symmetry lines in the Brillouin zone. MHM offers a rich tapestry of HOTI
and other topological and non-topological phases.
