  We here present a new version of the publicly available general relativistic
magnetohydrodynamic (GRMHD) code $\texttt{Spritz}$, which now includes an
approximate neutrino leakage scheme able to handle neutrino cooling and
heating. The leakage scheme is based on the publicly available
$\texttt{ZelmaniLeak}$ code, with a few modifications in order to properly work
with $\texttt{Spritz}$. We discuss the involved equations, physical
assumptions, and implemented numerical methods, along with a large battery of
general relativistic tests performed with and without magnetic fields. Our
tests demonstrate the correct implementation of the neutrino leakage scheme,
paving the way for further improvements of our neutrino treatment and the first
application to magnetized binary neutron star mergers. We also discuss the
implementation in the $\texttt{Spritz}$ code of high-order methods for a more
accurate evolution of hydrodynamical quantities.
