  We compute mixed QCD-weak corrections to inclusive Higgs production at the
LHC from the partonic process $g g \rightarrow H q \bar{q}$. We start from the
UV- and IR-finite one-loop weak amplitude and consider its interference with
the corresponding one-loop QCD amplitude. This contribution is a
$\mathcal{O}(\alpha_s\alpha)$ correction to the leading-order gluon-fusion
cross section, and was not numerically assessed in previous works. We also
compute the cross section from the square of this weak amplitude, suppressed by
$\mathcal{O}(\alpha^2)$. Finally, we consider contributions from the partonic
process $g q \rightarrow H q$, which are one order lower in $\alpha_s$, as a
reference for the size of terms which are not enhanced by the large gluon
luminosity. We find that, given the magnitude of the uncertainties on current
state-of-the-art predictions for Higgs production, all contributions computed
in this work can be safely ignored, both fully inclusively and in the boosted
Higgs regime. This result supports the approximate factorisation of QCD and
weak corrections to that process.
