Obstructive sleep apnoea (OSA) is a prevalent condition with significant health consequences, yet many patients remain undiagnosed due to the complexity and cost of over-night polysomnography. Acoustic-based screening provides a scalable alternative, yet performance is limited by environmental noise and the lack of physiological context. Respiratory effort is a key signal used in clinical scoring of OSA events, but current approaches require additional contact sensors that reduce scalability and patient comfort. This paper presents the first study to estimate respiratory effort directly from nocturnal audio, enabling physiological context to be recovered from sound alone. We propose a latent-space fusion framework that integrates the estimated effort embeddings with acoustic features for OSA detection. Using a dataset of 157 nights from 103 participants recorded in home environments, our respiratory effort estimator achieves a concordance correlation coefficient of 0.48, capturing meaningful respiratory dynamics. Fusing effort and audio improves sensitivity and AUC over audio-only baselines, especially at low apnoea-hypopnoea index thresholds. The proposed approach requires only smartphone audio at test time, which enables sensor-free, scalable, and longitudinal OSA monitoring.