This paper considers a sparse precoding design for sum-rate maximization in a cloud radio access network (Cloud-RAN). Constrained by the fronthaul link capacity and transmit power limit at each remote radio head (RRH), the sparse design amounts to determine the precoders at the RRHs as well as the set of serving RRHs for each mobile user. In this work, we first formulate the fronthaul link constraints as non-convex and discontinuous constraints with sparsity terms. These sparsity terms are then iteratively approximated into linear forms by means of reweighted ℓ1-norm with conjugate functions. Finally, to determine the beamforming vectors, the non-convex sum-rate maximization problem with linear constraints is transformed into an equivalent problem of iterative weighted mean-squared error minimization. Convergence of the proposed iterative algorithm is then proved and verified by the presented numerical results. In addition, numerical results demonstrate the superior performance by the proposed algorithm over a previously proposed one in literature.