In the Internet of Things (IoT) employing centralized machine learning, security is a major concern due to the heterogeneity of end devices. Malicious devices could launch poisoning attacks to degrade machine learning models. Distributed machine learning (DML) with blockchain provides a potential solution. Once local weights are recorded on the blockchain, model aggregation with defensive schemes can be executed on smartphones to prevent attacks. However, blockchain with the proof-of-work (PoW) consensus mechanism wastes computing resources and adds latency to DML. Computing resources can be utilized more efficiently with proof-of-useful-work (uPoW), which secures transactions by solving relevant real-world problems. We propose a novel uPoW method to minimize per-round latency of DML. The uPoW mining process schedules DML instances among multi-access edge computing (MEC) servers by solving a multi-way number partitioning problem. Moreover, poisoning attacks on heterogeneous training data pose significant challenges to blockchain-based DML. To address this problem, we propose a novel aggregation protocol, named ${Corrected\ Krum}$, to counter such attacks and improve the convergence speed of DML. By leveraging the mean-field approximation method, training errors are corrected to reduce the negative impact of poisoning attacks. Simulation results show that our proposed blockchain approach can significantly speed up DML compared with benchmarks.