  We have studied theoretically the Weyl semimetals the point symmetry group of
which has reflection planes and which contain equivalent valleys with opposite
chiralities. These include the most frequently studied compounds, namely the
transition metals monopnictides TaAs, NbAs, TaP, NbP, and also Bi$_{1-x}$Sb$_x$
alloys. The circular photogalvanic current, which inverts its direction under
reversal of the light circular polarization, has been calculated for the light
absorption under direct optical transitions near the Weyl points. In the
studied materials, the total contribution of all the valleys to the
photocurrent is nonzero only beyond the simple Weyl model, namely, if the
effective electron Hamiltonian is extended to contain either an anisotropic
spin-dependent linear contribution together with a spin-independent tilt or a
spin-dependent contribution cubic in the electron wave vector $\bf{k}$. With
allowance for the tilt of the energy dispersion cone in a Weyl semimetal of the
$C_{4v}$ symmetry, the photogalvanic current is expressed in terms of the
components of the second-rank symmetric tensor that determines the energy
spectrum of the carriers near the Weyl node; at low temperature, this
contribution to the photocurrent is generated within a certain limited
frequency range $\Delta $. The photocurrent due to the cubic corrections, in
the optical absorption region, is proportional to the light frequency squared
and generated both inside and outside the $\Delta$ window.
