  We have calculated the dynamical optical conductivity for
$\alpha-\mathcal{T}_3$ materials in the presence of a finite bandgap in their
energy bandstructure. This is a special type of energy dispersions because for
all $\alpha-\mathcal{T}_3$ materials with a bandgap, except graphene and a dice
lattice limits, the flat band receives a non-zero dispersion and assumes a
curved shape. The infinite ${\bf k}$-degeneracy of the flat energy band is also
lifted. Such a low-energy bandstructure could be obtained if an
$\alpha-\mathcal{T}_3$ material is irradiated off-resonant with circularly
polarized light. We have calculated the optical conductivity for the zero and
finite temperatures, as well as for the cases of a finite and nearly-zero
doping. We have demonstrated that analytical expressions could be in principle
obtained for all types of gapped $\alpha-\mathcal{T}_3$ materials and provided
the closed-form analytical expressions for a gapped dice lattice. Our numerical
results reveal some well-known signatures of the optical conductivity in
$\alpha-\mathcal{T}_3$ and silicene with two non-equivalent bandgaps, as well
as demonstrate some very specific features which have not been previously found
in any existing Dirac materials.
