We propose an algorithm for designing sparsity-constrained controllers for linear time-invariant systems with model uncertainties using a mixed $H_{2}/H_{\infty}$ objective. We first improve a previously proposed descent algorithm for mixed $H_{2}/H_{\infty}$ control using a modified Zoutendijk's method. Thereafter, we impose a sparsity constraint on this design by combining it with a greedy gradient support pursuit (GraSP) method. The proposed algorithm guarantees a predetermined level of sparsity while maintaining acceptable $H_{2}$ performance as well as $H_{\infty}$ robustness to uncertainties in the open-loop model. The effectiveness of the proposed method is illustrated through simulations.