  The crucial parameter of the current Monte Carlo models of high energy
hadron-hadron interaction is the transverse momentum cutoff $p_{T0}$ for
parton-parton interactions which slowly grows with energy and regularizes the
cross section. This modification of the collinear factorization formula goes
beyond the leading power and thus a natural question arises if such cutoff can
be extracted from a formalism which takes into account power corrections. In
this work, we consider the High Energy Factorization (HEF) valid at small $x$
and a new model, based on a similar principle to HEF, which in addition has a
limit respecting the Dokshitzer-Dyakonov-Troyan formula for the dijet momentum
disbalance spectrum. Minijet cross section and its suppression is then analyzed
in two ways. First, we study minijets directly in the low-$p_T$ region, and
demonstrate that higher twist corrections do generate suppression of the
inclusive jet production cross section though these effects are not leading to
the increase of the cutoff with incident energy. Second, we consider hard
inclusive dijet production where Multi Parton Interactions (MPIs) with minijets
produce power corrections. We introduce an observable constructed from
differential cross section in the ratio $\tau$ of dijet disbalance to the
average dijet $p_T$ and demonstrate that the $\tau>1$ region is sensitive to
the cutoff $p_{T0}$ in the MPI minijet models. The energy dependence of the
cutoff is reflected in the energy dependence of the bimodality coefficient $b$
of the $\tau$ distribution. We compare $b$ calculated from $\mathsf{pythia}$,
where one can conveniently control MPIs by the program parameters, and HEF for
a few unintegrated gluon distributions (UGDs). We find that the energy
dependence of $b$ is very sensitive to the particular choice of UGD and in some
models it resembles predictions of the Monte Carlo models.
