  We present an experimental and numerical study of three-magnon splitting in a
micrometer-sized magnetic disk with the vortex state strongly deformed by
static in-plane magnetic fields. Excited with a large enough power at frequency
$f_\mathrm{RF}$, the primary radial magnon modes of a cylindrical magnetic
vortex can decay into secondary azimuthal modes via spontaneous three-magnon
splitting. This nonlinear process exhibits selection rules leading to
well-defined and distinct frequencies $f_\mathrm{RF}/2\pm \Delta f$ of the
secondary modes. Here, we demonstrate that three-magnon splitting in vortices
can be significantly modified by deforming the magnetic vortex with in-plane
magnetic fields, leading to a much richer three-magnon response. We find that,
with increasing field, an additional class of secondary modes is excited which
are localized to the highly-flexed regions adjacent to the displaced vortex
core. While these modes satisfy the same selection rules of three-magnon
splitting, they exhibit a much lower three-magnon threshold power compared to
regular secondary modes of a centered vortex. The applied static magnetic
fields are small ($\simeq$ 10 mT), providing an effective parameter to control
the nonlinear spectral response of confined vortices. Our work expands the
understanding of nonlinear magnon dynamics in vortices and advertises these for
potential neuromorphic applications based on magnons.
