Constructing large, labeled training datasets for segmentation models is an expensive and labor-intensive process. This is a common challenge in machine learning, addressed by methods that require few or no labeled data points such as few-shot learning (FSL) and weakly-supervised learning (WS). Such techniques, however, have limitations when applied to image segmentation---FSL methods often produce noisy results and are strongly dependent on which few datapoints are labeled, while WS models struggle to fully exploit rich image information. We propose a framework that fuses FSL and WS for segmentation tasks, enabling users to train high-performing segmentation networks with very few hand-labeled training points. We use FSL models as weak sources in a WS framework, requiring a very small set of reference labeled images, and introduce a new WS model that focuses on key areas---areas with contention among noisy labels---of the image to fuse these weak sources. Empirically, we evaluate our proposed approach over seven well-motivated segmentation tasks. We show that our methods can achieve within 1.4 Dice points compared to fully supervised networks while only requiring five hand-labeled training points. Compared to existing FSL methods, our approach improves performance by a mean 3.6 Dice points over the next-best method.