  We theoretically study the impact of impurities on the photogalvanic effect
(PGE) in Weyl semimetals with weakly tilted Weyl cones. Our calculations are
based on a two-nodes model with an inversion symmetry breaking offset and we
employ a kinetic equation approach in which both optical transitions as well as
particle-hole excitations near the Fermi energy can be taken into account. We
focus on the parameter regime with a single photoactive node and control the
calculation in small impurity concentration. Internode scattering is treated
generically and therefore our results allow to continuously interpolate between
the cases of short range and long range impurities. We find that the time
evolution of the circular PGE may be nonmonotonic for intermediate internode
scattering. Furthermore, we show that the tilt vector introduces three
additional linearly independent components to the steady state photocurrent.
Amongst them, the photocurrent in direction of the tilt takes a particular role
inasmuch it requires elastic internode scattering or inelastic intranode
scattering to be relaxed. It may therefore be dominant. The tilt also generates
skew scattering which leads to a current component perpendicular to both the
incident light and the tilt. We extensively discuss our findings and comment on
the possible experimental implications.
