  Background field methods provide an important nonperturbative formalism for
the determination of hadronic properties which are complementary to
matrix-element calculations. However, new challenges are encountered when
utilising a fermion action exposed to additive mass renormalisations. In this
case, the background field can induce an undesired field-dependent additive
mass renormalisation that acts to change the quark mass as the background field
is changed. For example, in a calculation utilising Wilson fermions in a
uniform background magnetic field, the Wilson term introduced a field-dependent
renormalisation to the quark mass which manifests itself in an unphysical
increase of the neutral-pion mass for large magnetic fields. Herein, the clover
fermion action is studied to determine the extent to which the removal of
$\mathcal{O}(a)$ discretisation errors suppresses the field-dependent changes
to the quark mass. We illustrate how a careful treatment of nonperturbative
improvement is necessary to resolve this artefact of the Wilson term. Using the
$32^3 \times 64$ dynamical-fermion lattices provided by the PACS-CS
Collaboration we demonstrate how our technique suppresses the unphysical mass
renormalisation over a broad range of magnetic field strengths.
