  The study of the Higgs boson properties is one of the main tasks of
contemporary high-energy physics. Among Higgs properties, its interaction with
gluons is interesting since it can be facilitated by yet unknown elementary
particles. One of the major sources of uncertainty in the theoretical
description of $ggH$ coupling originates from mixed QCD-electroweak
contributions. The NLO QCD corrections to these contributions were evaluated in
the approximation where electroweak boson masses were considered to be
significantly larger than the mass of the Higgs boson and it is desirable to
compute these corrections for physical masses of the gauge bosons and the Higgs
boson. We present a major step towards this goal and describe first the
analytic evaluation of NLO mixed QCD-EW three-loop virtual corrections to $gg
\to H$, and then their implementation in the evaluation of the total cross
section for $gg \to H$ in the soft-gluon approximation for real corrections.
