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 band gap. 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 band gap. 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 thresholdlike behavior due to the dynamical gap opening and renormalizations of the density of states and velocity of quasiparticles.