Modern medicine relies on heterogeneous data sources spanning radiology, pathology, text reports, and structured clinical information. However, real-world patient data are frequently incomplete, with missing or sparsely acquired modalities, limiting the effectiveness of standard multimodal fusion approaches. To this end, we propose the Multimodal Flexible Redundancy-aware decomposed GAted Learning (Multi-FRuGaL) framework, a decomposition-aware, adaptive gated intermediate-fusion framework that performs modality-level representation learning under missing data. Multi-FRuGaL integrates per-modality encoders with a signal decomposition layer, an input-conditioned gating network, and an information-aware fusion objective to separate redundant from modality-specific complementary signals, selectively upweighting informative modalities and suppressing redundant or noisy inputs, and remaining well-defined even when multiple modalities are absent. We evaluate Multi-FRuGaL on two multimodal head and neck cancer cohorts: the HANCOCK challenge dataset (N = 763) comprising five modalities and two prognostic endpoints (5-year survival and 2-year recurrence), and the HECKTOR challenge dataset (N = 588) comprising three modalities for human papillomavirus (HPV) status classification. Multi-FRuGaL consistently achieves higher mean performance than the evaluated baselines across multiple tasks, improving AUC from 0.601 to 0.8496 for survival, from 0.672 to 0.8102 for recurrence, and achieving 0.975 AUC for HPV prediction on HECKTOR. For survival analysis, it further achieves a concordance index of 0.6814 for overall survival, 0.7421 for recurrence-free survival, and 0.7143 for progression-free survival on HANCOCK, and 0.7203 for recurrence-free survival on HECKTOR. Qualitative analyses further show that Multi-FRuGaL learns discriminative and robust multimodal representations, even under severe missing-modality conditions.