  We study topological properties and the topological phase transitions therein
for a semi-Dirac Haldane model on a honeycomb lattice in presence of an
extended range (third neighbour) hopping. While in the absence of a third
neighbour hopping, $t_3$, the system exhibits gapless electronic spectrum, its
presence creates an energy gap in the dispersion. However, the nature of the
spectral gap, that is, whether it is trivial or topological needs to be
ascertained. We find that the answer depends on the value of $t_3$, and its
interplay with the value of the onsite potential that breaks the sublattice
symmetry, namely, Semenoff mass ($\Delta$). To elucidate our findings on the
topological phases, we demonstrate two kinds of phase diagrams using the
available parameter space, one in which the phases are shown in the
$\Delta$-$t_3$ plane, and the other one in a more familiar $\Delta$-$\phi$
plane ($\phi$ being the Haldane flux). The phase diagrams depict the presence
of Chern insulating lobes comprising of Chern numbers $\pm2$ and $\pm1$ for
certain values of $t_3$, along with trivial insulating regions (zero Chern
number). Thus there are phase transitions from one topological regime to
another which are characterized by abrupt changes in the values of the Chern
number. To support the existence of the topological phases, we compute the
counter-propagating chiral edge modes in a ribbon geometry. Finally, the
anomalous Hall conductivity shows plateaus either at $e^2/h$ or $2e^2/h$
corresponding to these topological phases.
