  Radiatively generated neutrino masses ($m_\nu$) are proportional to
supersymmetry (SUSY) breaking, as a result of the SUSY non-renormalisation
theorem. In this work, we investigate the space of SUSY radiative seesaw models
with regard to their dependence on SUSY breaking
($\require{cancel}\cancel{\text{SUSY}}$). In addition to contributions from
sources of $\cancel{\text{SUSY}}$ that are involved in electroweak symmetry
breaking ($\cancel{\text{SUSY}}_\text{EWSB}$ contributions), and which are
manifest from $\langle F^\dagger_H \rangle = \mu \langle \bar H \rangle \neq 0$
and $\langle D \rangle = g \sum_H \langle H^\dagger \otimes_H H \rangle \neq
0$, radiatively generated $m_\nu$ can also receive contributions from
$\cancel{\text{SUSY}}$ sources that are unrelated to EWSB
($\cancel{\text{SUSY}}_\text{EWS}$ contributions). We point out that recent
literature overlooks pure-$\cancel{\text{SUSY}}_\text{EWSB}$ contributions
($\propto \mu / M$) that can arise at the same order of perturbation theory as
the leading order contribution from $\cancel{\text{SUSY}}_\text{EWS}$.
  We show that there exist realistic radiative seesaw models in which the
leading order contribution to $m_\nu$ is proportional to
$\cancel{\text{SUSY}}_\text{EWS}$. To our knowledge no model with such a
feature exists in the literature. We give a complete description of the
simplest model-topologies and their leading dependence on
$\cancel{\text{SUSY}}$. We show that in one-loop realisations $L L H H$
operators are suppressed by at least $\mu \, m_\text{soft} / M^3$ or
$m_\text{soft}^2 / M^3$. We construct a model example based on a one-loop
type-II seesaw. An interesting aspect of these models lies in the fact that the
scale of soft-$\cancel{\text{SUSY}}$ effects generating the leading order
$m_\nu$ can be quite small without conflicting with lower limits on the mass of
new particles.
