Hospital-acquired infections (HAIs), particularly those caused by multidrug-resistant (MDR) bacteria, pose significant risks to vulnerable patients. Accurate predictive models are important for assessing infection dynamics and informing infection prediction and control (IPC) programmes. Graph-based methods, including graph neural networks (GNNs), offer a powerful approach to model complex relationships between patients and environments but often struggle with data sparsity, irregularity, and heterogeneity. We propose the space-time-and-memory (STM)-GNN, a temporal GNN enhanced with recurrent connectivity designed to capture spatiotemporal infection dynamics. STM-GNN effectively integrates sparse, heterogeneous data combining network information from patient-environment interactions and internal memory from historical colonization and contact patterns. Using a unique IPC dataset containing clinical and environmental colonization information collected from a long-term healthcare unit, we show that STM-GNN effectively addresses the challenges of limited and irregular data in an MDR prediction task. Our model reaches 0.84 AUROC, and achieves the most balanced performance overall compared to classic machine learning algorithms, as well as temporal GNN approaches.