We here present a new version of the publicly available general relativistic magnetohydrodynamic (GRMHD) code 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 ZelmaniLeak code, with a few modifications in order to properly work with 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 Spritz code of high-order methods for a more accurate evolution of hydrodynamical quantities.