The stochastic heat equation on the sphere driven by an additive square-integra\-ble Lévy process is approximated by a spectral method in space and forward and backward Euler--Maruyama schemes in time. New regularity results are proven for its solution. The spectral approximation is based on a truncation of the series expansion with respect to the spherical harmonic functions. For a given regularity of the initial condition and two different settings of regularity for the driving noise, strong convergence rates for the spectral approximation and for the Euler--Maruyama methods are proven. Moreover, weak rates of up to twice the strong rates are shown. Numerical simulations confirm the theoretical results.