  We study the associated production of an electroweak vector boson and the
Higgs boson with a jet via gluon-gluon fusion. At the leading order, these
processes occur at one-loop level. The amplitudes of these one-loop processes
are gauge invariant and finite. Therefore, their contributions towards the
corresponding hadronic cross sections and kinematic distributions can be
calculated separately. We present results for the Large Hadron Collider and its
discussed upgrades. We find that the gluon-gluon one-loop process gives
dominant contribution to the $\gamma H j$ production. We observe a destructive
interference effect in the $gg\to Z H j$ amplitude. We also find that in the
high transverse momentum and central rapidity region, the $ZHj$ production
cross section via gluon-gluon fusion becomes comparable to the cross section
contributions coming from quark-quark and quark-gluon channels.
