  We present electronic and transport properties of a zigzag nanoribbon made of
alpha-$\mathcal{T}_3$ lattice. Our particular focus is on the effects of the
continous evolution of the edge modes ( from flat to dispersive) on the
thermoelectric transport properties. Unlike the case of graphene nanoribbon,
the zigzag nanoribbon of $\alpha-\mathcal{T}_3$ lattice can host a pair of
dispersive (chiral) edge modes at the two valleys for specific width of the
ribbon. Moreover, gap opening can also occur at the two valleys depending on
the width. The slope of the chiral edge modes and the energy gap strongly
depend on the relative strength of two kinds of hoping parameters present in
the system. We compute corresponding transport coefficients such as
conductance, thermopower, thermalconductivity and the thermoelectric figure of
merits by using the tight-binding Green function formalism, in order to explore
the roles of the dispersive edge modes. It is found that the thermopower and
thermoelectric figure of merits can be enhanced significantly by suitably
controlling the edge modes. The figure of merits can be enhanced by ten times
under suitable parameter regime in comparison to the case of graphene. Finally,
we reveal that the presence of line defect, close to the edge, can cause a
significant impact on the edge modes as well as on electrical conductance.
However, thermopower is relatively less sensitive to such defects.
