<p>Over the past few years, there has been a growing utilization of deep neural networks (DNN) for a range of intricate tasks in medical image analysis (MIA), such as disease diagnosis, patient categorization, and aiding clinical decisions. Conventionally, training cognitive computation networks like DNNs in fully supervised tasks requires a considerable volume of well-annotated, quality data to ensure effective learning and prediction performance. Furthermore, more often than not, obtaining high-quality annotations in the medical field for training DNN is a predominant challenge. Consequently, it becomes crucial to devise cognitive computation algorithms tailored for training DNN using sparsely labeled data while ensuring strong generalization capabilities. We identified a notable scarcity of user studies and surveys investigating approaches and strategies for working with insufficient data and annotations in the MIA field, especially for image segmentation. This work aims to address this gap. Following the PRISMA framework, a total of 104 peer-reviewed studies published between 2018 and 2025 are included in this work, primarily sourced from the Scopus and the Google scholar databases, based on predefined inclusion and exclusion criteria, ensuring relevance to data and annotation-efficient medical image segmentation. Unlike the previous review papers, this survey extends beyond traditional and deep learning-based augmentation techniques or deep semi-supervised approaches, by explicitly focusing on medical/clinical imaging modalities, such as CT, MRI, and X-ray, offering a broader perspective, with techniques broadly categorized under the domains of Data Augmentation, Transfer Learning, Unsupervised Domain Adaptation, Knowledge Distillation, Federated Learning, Zero-Shot Learning, One-Shot Learning, Few-Shot Learning, Weakly Supervised Learning, and Active Learning.</p>