We propose a fast algorithm for the reconstruction of non-cartesian acquisitions with B0-inhomogeneity. The proposed method uses a new SVD-based approximation of the B0-aware imaging operator in combination with diagonal k-space preconditioning. The proposed SVD-based approximation adaptively determines the required number of basis functions and thus reduces the computational effort. Furthermore, we present a method to efficiently compute the $$$\ell_2$$$-optimal diagonal k-space preconditioner taking into account the B0-map. The obtained preconditioner closely matches the one without B0-map. Our experiments demonstrate the fast convergence and reduced computational costs of the proposed method.