  An earlier developed $k_T$-factorization framework to calculate the
associated prompt photon and hadronic jets production cross sections at high
energies is extended now to the electron-proton deep inelastic scattering. The
proposed method is based on joint usage of the PEGASUS and CASCADE Monte-Carlo
event generators, which deal with the transverse momentum dependent (TMD) gluon
and quark densities in a proton. First of them, PEGASUS, is applied to produce
off-shell photon-gluon fusion events. Then, to properly simulate the kinematics
of the produced jets, the TMD parton shower algorithm implemented into the
CASCADE is used. We demonstrate basic features of the approach considering
latest H1 and ZEUS data on prompt photon plus jet production at low $Q^2$
collected at HERA. We achieve a good description of the measurements and point
out that a correct simulation of parton showers is essential for studying
non-inclusive processes at high energies within the TMD-based approaches.
