Based on a multi-agent system model with random delay and random communication noise under a switched network, a distributed multi-step approximate subgradient random projection algorithm is proposed, and the algorithm convergence analysis is performed. Firstly, we convert a network with random communication delay into a network without delay by using a network expansion method. Then, we propose the concept of approximate subgradient, and design a multi-step approximate subgradient batch random projection algorithm. A batch random projection method is used to deal with the executive problem of projection operators when the overall constraint set is not easy to obtain in practical problems. Finally, the numerical simulations show that the proposed algorithm has better convergence effects than the general distributed multi-step subgradient algorithm even if there exists random noise. And the effect of the number of random projection sets and random noise are also discussed.