  Using micromagnetic simulations, we study the interplay between strongly
voltage-controlled magnetic anisotropy (VCMA), $\Delta K = \pm$200 kJ/m$^3$,
and gate width, $w=$ 10--400 nm, in voltage-gated W/CoFeB/MgO based
nano-constriction spin Hall nano-oscillators. The VCMA modifies the local
magnetic properties such that the magnetodynamics transitions between regimes
of \emph{i}) confinement, \emph{ii}) tuning, and \emph{iii}) separation, with
qualitatively different behavior. We find that the strongest tuning is achieved
for gate widths of the same size as the the constriction width, for which the
effective damping can be increased an order of magnitude compared to its
intrinsic value. As a consequence, voltage control remains efficient over a
very large frequency range, and subsequent manufacturing advances could allow
SHNOs to be easily integrated into next-generation electronics for further
fundamental studies and industrial applications.
