  We consider the associated production of Higgs boson and hadronic jet(s) in
$pp$ collisions at the LHC for the first time using the $k_T$-factorization
approach. Our analysis is based on the off-shell gluon-gluon fusion subprocess,
where non-zero transverse momenta of initial gluons are taken into account and
covers $H \to \gamma \gamma$, $H \to ZZ^* \to 4l$ (with $l = e,\mu$) and $H \to
W^+ W^- \to e^\pm \mu^\mp \nu \bar \nu$ decay channels. The transverse momentum
dependent (TMD) gluon densities in a proton are taken from
Catani-Ciafaloni-Fiorani-Marchesini evolution equation. To simulate the
kinematics of the produced jets the TMD parton shower implemented into the
Monte-Carlo event generator \textsc{cascade} is applied. The comparison of our
results with the latest experimental data taken by the CMS and ATLAS
Collaborations at $\sqrt s = 8$ and $13$ TeV and conventional higher-order
perturbative QCD calculations is presented. We highlight observables, which are
sensitive to the TMD gluon densities in a proton.
