Federated learning offers a promising paradigm for training machine learning models in distributed edge networks. This paper focuses on a critical aspect of federated learning, specifically examining the influence of communication delays between the server and edge devices. We study the problem of federated composite optimization (FCO) in the scenario where communication delays are introduced due to client-sides. To tackle this issue, we propose a federated dual averaging learning algorithm that incorporates delayed gradients. We conduct a thorough convergence analysis of our algorithm taking into account the impact of the delayed gradient information. Our analysis demonstrates that under the assumption of bounded expected random delays, the algorithm exhibits the asymptotic convergence towards the optimal value. Furthermore, we present explicit convergence rates of the proposed algorithm for fixed and random delays. Our results show that the impact of delays on FCO problems with smooth loss functions gradually becomes negligible. Finally, to demonstrate the efficacy of the proposed algorithm, we conduct numerical simulations on the FEMNIST dataset utilizing ℓ1-regularized logistic regression. These experimental results confirm our findings that the algorithm converges effectively even in the presence of delays.