  Recent analyses on high-energy inclusive Higgs-boson rates in proton
collisions via the gluon-fusion channel, matched with the state of-the-art
fixed-order N$^3$LO accuracy, have shown that the impact of high-energy
resummation corrections reaches 10% at the FCC nominal energies. This supports
the statement that electroweak physics at 100 TeV is expected to receive
relevant contributions from small-$x$ physics. In this preliminary study we
present novel predictions for transverse-momentum and rapidity distributions
sensitive to the inclusive emission of a Higgs boson in association with a
light-flavored jet in proton collisions, calculated within the NLL accuracy of
the energy-logarithmic resummation. We highlight how high-energy signals for
this process are already present and visible at current LHC energies, and they
are also sizable at the FCC ones. We come out with the message that the
improvement of fixed-order calculations on Higgs-sensitive QCD distributions is
a core ingredient to reach the precision level in the description of
observables relevant for the Higgs physics at the FCC.
