  We propose and study generic multi-Weyl semimetal (mWSM) lattice Hamiltonians
that break particle-hole symmetry. These models fall into two categories: model
I (model II) where the gap and tilt terms are coupled (decoupled) can host
type-I and type-II Weyl nodes simultaneously (separately) in a hybrid phase
(type-I and type-II phases, respectively). We concentrate on the question of
how anisotropy and non-linearity in the dispersions, gaps and tilt terms
influence diffusive second order transport quantities namely, the circular
photogalvanic effect (CPGE) and the Berry curvature dipole (BCD) as well as
first order Magnus Hall effect (MHE) in the ballistic limit. The signatures of
topological charges are clearly imprinted in the quantized CPGE response for
the hybrid mWSM phase in model I. Such a quantization is also found in the
type-I WSM phase for model II, however, the frequency profiles of the CPGE in
these two cases is distinctively different owing to their different band
dispersion irrespective of the identical topological properties. The
contributions from the vicinity of Weyl nodes and away from the WNs are clearly
manifested in the BCD response, respectively, for model I model II. The Fermi
surface properties for the activated momentum lead to a few hallmark features
on the MHE for both the models. Furthermore, we identify distinguishing
signatures of the above responses for type-I, type-II and hybrid phases to
provide an experimentally viable probe to differentiate these WSMs phases.
