  Electromagnetic dressing by a high-frequency field drastically modifies the
electronic transport properties on Dirac systems. Here its effects on the
energy spectrum of graphene with two possible phases of Kekul\'e distortion
(namely, Kek-Y and Kek-O textures) are studied. Using Floquet theory it is
shown how circularly polarized light modifies the gapless spectrum of the Kek-Y
texture, producing dynamical band gaps at the Dirac point that depends on the
amplitude and the frequency of the electric field, and breaks the valley
degeneracy of the gapped spectrum of the Kek-O texture. To further explore the
electronic properties under circularly polarized radiation, the dc conductivity
is studied by using the Boltzmann approach and considering both inter-valley
and intra-valley contributions. When linearly polarized light is considered,
the band structure of both textures is always modified in a perpendicular
direction to the electric field. While the band structure for the Kek-Y texture
remains gapless, the gap for the Kek-O texture is reduced considerably. For
this linear polarization it is also shown that non-dispersive bands can appear
by a precise tuning of the light field parameters thus inducing dynamical
localization. The present results suggest that optical measurements will allow
to distinguish between different Kekul\'e bond textures.
