The issue of domain shift remains a problematic phenomenon in most real-world datasets and clinical audio is no exception. In this work, we study the nature of domain shift in a clinical database of infant cry sounds acquired across different geographies. We find that though the pitches of infant cries are similarly distributed regardless of the place of birth, other characteristics introduce peculiar biases into the data. We explore methodologies for mitigating the impact of domain shift in a model for identifying neurological injury from cry sounds. We adapt unsupervised domain adaptation methods from computer vision which learn an audio representation that is domain-invariant to hospitals and is task discriminative. We also propose a new approach, target noise injection (TNI), for unsupervised domain adaptation which requires neither labels nor training data from the target domain. Our best-performing model significantly improves target accuracy by 7.2%, without negatively affecting the source domain.