  The quantisation of scalar field theory and Einstein gravity is investigated
using a fully covariant background field formalism, including Vilkovisky-DeWitt
corrections. The one-loop divergences, which are relevant for the consistency
of the low-energy effective theory, differ substantially from non-covariant
calculations. The results are applied to the effective action of Higgs
inflation, with a non-minimal gravity coupling parameter $\xi$, where the use
of a formalism which is independent of field redefinitions is particularly
important. Consistency of the one-loop effective action requires the cut-off
$\Lambda<M_p/\sqrt{\xi}$ in the small field and $\Lambda<M_p$ in the large
field limit.
