Federated learning (FL) allows multiple edge computing nodes to jointly build a shared learning model without having to transfer their raw data to a centralized server, thus reducing communication overhead. However, FL still faces a number of challenges such as nonindependent and identically distributed data and heterogeneity of user equipments (UEs). Enabling a large number of UEs to join the training process in every round raises a potential issue of the heavy global communication burden. To address these issues, we generalize the current state-of-the-art federated averaging (FedAvg) by adding a weight-based proximal term to the local loss function. The proposed FL algorithm runs stochastic gradient descent in parallel on a sampled subset of the total UEs with replacement during each global round. We provide a convergence upper bound characterizing the tradeoff between convergence rate and global rounds, showing that a small number of active UEs per round still guarantees convergence. Next, we employ the proposed FL algorithm in wireless Internet-of-Things (IoT) networks to minimize either total energy consumption or completion time of FL, where a simple yet efficient path-following algorithm is developed for its solutions. Finally, numerical results on unbalanced data sets are provided to demonstrate the performance improvement and robustness on the convergence rate of the proposed FL algorithm over FedAvg. They also reveal that the proposed algorithm requires much less training time and energy consumption than the FL algorithm with full user participation. These observations advocate the proposed FL algorithm for a paradigm shift in bandwidth-constrained learning wireless IoT networks.