We are interested in the discretization of stable driven SDEs with additive noise for $α$ $\in$ (1, 2) and Lq -- Lp drift under the Serrin type condition $α$/q + d/p < $α$ -- 1. We show weak existence and uniqueness as well as heat kernel estimates for the SDE and obtain a convergence rate of order (1/$α$)*($α$ -- 1 -- $α$/q - d/p) for the difference of the densities for the Euler scheme approximation involving suitably cutoffed and time randomized drifts.