  We develop a microscopic theory of an unconventional photogalvanic effect in
two-dimensional materials with the Dirac energy spectrum of the carriers of
charge under strong driving. As a test bed, we consider a layer of a transition
metal dichalcogenide, exposed to two different electromagnetic fields. The
first pumping field is circularly-polarized, and its frequency exceeds the
material bandgap. It creates an extremely nonequilibrium distribution of
electrons and holes in one valley (K) and opens dynamical gaps, whereas the
other valley (K') remains empty due to the valley-dependent interband selection
rules. The second probe field has the frequency much smaller than the material
bandgap. It generates intraband perturbations of the nonequilibrium carriers
density, resulting in the photogalvanic current due to the trigonal asymmetry
of the dispersions. This current shows threshold-like behavior due to the
dynamical gap opening and renormalizations of the density of states and
velocity of quasiparticles.
