  We report progress on the study of the inclusive semi-hard hadroproduction of
a Higgs+jet system at LHC and FCC collision energies. We describe a prototype
matching procedure aimed at combining NLO fixed-order computations via POWHEG,
with the NLL resummation of energy logarithms from JETHAD. We present
preliminary analyses on assessing the weight of systematic uncertainties, such
as the ones coming from finite top- and bottom-quark masses.
