  Normal-mode oscillation frequencies computed from stellar models differ from
those which would be measured from stars with identical interior structures,
because of modelling errors in the near-surface layers. These frequency
differences are referred to as the asteroseismic "surface term". The vast
majority of solar-like oscillators which have been observed, and which are
expected to be observed in the near future, are evolved stars which exhibit
mixed modes. For these evolved stars, the inference of stellar properties from
these mode frequencies has been shown to depend on how this surface term is
corrected for. We show that existing parametrisations of the surface term
account for mode mixing only to first order in perturbation theory, if at all,
and therefore may not be adequate for evolved stars. Moreover, existing
nonparametric treatments of the surface term do not account for mode mixing. We
derive both a first-order construction, and a more general approach, for one
particular class of nonparametric methods. We illustrate the limits of
first-order approximations from both analytic considerations and using
numerical injection-recovery tests on stellar models. First-order corrections
for the surface term are strictly only applicable where the size of the surface
term is much smaller than both the coupling strength between the mixed p- and
g-modes, as well as the local g-mode spacing. Our more general matrix
construction may be applied to evolved stars, where perturbation theory cannot
be relied upon.
