  The Euclid space telescope will survey a large dataset of cosmic voids traced
by dense samples of galaxies. In this work we estimate its expected performance
when exploiting angular photometric void clustering, galaxy weak lensing and
their cross-correlation. To this aim, we implement a Fisher matrix approach
tailored for voids from the Euclid photometric dataset and present the first
forecasts on cosmological parameters that include the void-lensing correlation.
We examine two different probe settings, pessimistic and optimistic, both for
void clustering and galaxy lensing. We carry out forecast analyses in four
model cosmologies, accounting for a varying total neutrino mass, $M_\nu$, and a
dynamical dark energy (DE) equation of state, $w(z)$, described by the CPL
parametrisation. We find that void clustering constraints on $h$ and $\Omega_b$
are competitive with galaxy lensing alone, while errors on $n_s$ decrease
thanks to the orthogonality of the two probes in the 2D-projected parameter
space. We also note that, as a whole, the inclusion of the void-lensing
cross-correlation signal improves parameter constraints by $10-15\%$, and
enhances the joint void clustering and galaxy lensing Figure of Merit (FoM) by
$10\%$ and $25\%$, in the pessimistic and optimistic scenarios, respectively.
Finally, when further combining with the spectroscopic galaxy clustering,
assumed as an independent probe, we find that, in the most competitive case,
the FoM increases by a factor of 4 with respect to the combination of weak
lensing and spectroscopic galaxy clustering taken as independent probes. The
forecasts presented in this work show that photometric void-clustering and its
cross-correlation with galaxy lensing deserve to be exploited in the data
analysis of the Euclid galaxy survey and promise to improve its constraining
power, especially on $h$, $\Omega_b$, the neutrino mass, and the DE evolution.
