  Massive neutrinos and $f(R)$ modified gravity have degenerate observational
signatures that can impact the interpretation of results in galaxy survey
experiments, such as cosmological parameter estimations and gravity model
tests. Because of this, it is important to investigate astrophysical
observables that can break these degeneracies. Cosmic voids are sensitive to
both massive neutrinos and modifications of gravity and provide a promising
ground for disentangling the above mentioned degeneracies. In order to analyse
cosmic voids in the context of non-$\Lambda$CDM cosmologies, we must first
understand how well the current theoretical framework operates in these
settings. We performed a suite of simulations with the RAMSES-based N-body code
ANUBISIS, including massive neutrinos and $f(R)$ modified gravity both
individually and simultaneously. The data from the simulations were compared to
models of the void velocity profile and the void-halo cross-correlation
function (CCF). This was done both with the real space simulation data as model
input and by applying a reconstruction method to the redshift space data. In
addition, we ran Markov chain Monte Carlo (MCMC) fits on the data sets to
assess the capability of the models to reproduce the fiducial simulation values
of $f\sigma_8(z)$ and the Alcock-Paczy\`{n}ski parameter, $\epsilon$. The void
modelling applied performs similarly for all simulated cosmologies, indicating
that more accurate models and higher resolution simulations are needed in order
to directly observe the effects of massive neutrinos and $f(R)$ modified
gravity through studies of the void-galaxy CCF. The MCMC fits show that the
choice of void definition plays an important role in the recovery of the
correct cosmological parameters, but otherwise, there is no clear distinction
between the ability to reproduce $f\sigma_8$ and $\epsilon$ for the various
simulations.
