Estimating fish body measurements like length, width, and mass has received considerable research due to its potential in boosting productivity in marine and aquaculture applications. Some methods are based on manual collection of these measurements using tools like a ruler which is time consuming and labour intensive. Others rely on fully-supervised segmentation models to automatically acquire these measurements but require collecting per-pixel labels which are also time consuming. It can take up to 2 minutes per fish to acquire accurate segmentation labels. To address this problem, we propose a segmentation model that can efficiently train on images labeled with point-level supervision, where each fish is annotated with a single click. This labeling scheme takes an average of only 1 second per fish. Our model uses a fully convolutional neural network with one branch that outputs per-pixel scores and another that outputs an affinity matrix. These two outputs are aggregated using a random walk to get the final, refined per-pixel output. The whole model is trained end-to-end using the localization-based counting fully convolutional neural network (LCFCN) loss and thus we call our method Affinity-LCFCN (A-LCFCN). We conduct experiments on the DeepFish dataset, which contains several fish habitats from north-eastern Australia. The results show that A-LCFCN outperforms a fully-supervised segmentation model when the annotation budget is fixed. They also show that A-LCFCN achieves better segmentation results than LCFCN and a standard baseline.