  Cosmic voids are progressively emerging as a new viable cosmological probe.
Their abundance and density profiles are sensitive to modifications of gravity,
as well as to dark energy and neutrinos. The main goal of this work is to
investigate the possibility of exploiting cosmic void statistics to disentangle
the degeneracies resulting from a proper combination of $f(R)$ modified gravity
and neutrino mass. We use N-body simulations to analyse the density profiles
and size function of voids traced by both dark matter particles and haloes. We
find clear evidence of the enhancement of gravity in $f(R)$ cosmologies in the
void density profiles at $z=1$. However, these effects can be almost completely
overridden by the presence of massive neutrinos because of their thermal
free-streaming. Despite the limited volume of the analysed simulations does not
allow us to achieve a statistically relevant abundance of voids larger than $40
\ \mathrm{Mpc}/h$, we find that the void size function at high redshifts and
for large voids is potentially an effective probe to disentangle these
degenerate cosmological models, which is key in the prospective of the upcoming
wide field redshift surveys.
