We explore the topological properties of a modified Haldane model (MHM) in which the strength of the nearest-neighbor and next-nearest-neighbor hopping terms is made unequal and the threefold rotational symmetry C-3 is broken by introducing a dimerization term (vertical bar t(1w(2w))vertical bar < t(1s(2s))) in the Hamiltonian. Using the parameter eta = t(1w)/t(1s) = t(2w)/t(2s), we show that this MHM supports a transition from the quantum anomalous Hall insulator to a higher-order topological insulator (HOTI) phase at eta = +/- 0.5. It also hosts a zero-energy corner mode on a nanodisk 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. Our MHM offers a rich tapestry of HOTIs and other topological and nontopological phases.