  Observations of pressure-gravity mixed modes, combined with a theoretical
framework for understanding mode formation, can yield a wealth of information
about deep stellar interiors. In this paper, we seek to develop a formalism for
treating the effects of deeply buried core magnetic fields on mixed modes in
evolved stars, where the fields are moderate, i.e. not strong enough to disrupt
wave propagation, but where they may be too strong for non-degenerate
first-order perturbation theory to be applied. The magnetic field is
incorporated in a way that avoids having to use this. Inclusion of the Lorentz
force term is shown to yield a system of differential equations that allows for
the magnetically-affected eigenfunctions to be computed from scratch, rather
than following the approach of first-order perturbation theory. For
sufficiently weak fields, coupling between different spherical harmonics can be
neglected, allowing for reduction to a second-order system of ordinary
differential equations akin to the usual oscillation equations that can be
solved analogously. We derive expressions for (i) the mixed-mode quantisation
condition in the presence of a field and (ii) the frequency shift associated
with the magnetic field. In addition, for modes of low degree we uncover an
extra offset term in the quantisation condition that is sensitive to properties
of the evanescent zone. These expressions may be inverted to extract
information about the stellar structure and magnetic field from observational
data.
