Instance segmentation methods require large datasets with expensive instance-level mask labels. This makes partially supervised learning appealing in settings where abundant box and limited mask labels are available. To improve mask predictions with limited labels, we modify a Mask R-CNN by introducing an object mask prior (OMP) for the mask head. We show that a conventional class-agnostic mask head has difficulties learning foreground for classes with box-supervision only. Our OMP resolves this by providing the mask head with the general concept of foreground implicitly learned by the box classification head under the supervision of all classes. This helps the class-agnostic mask head to focus on the primary object in a region of interest (RoI) and improves generalization to novel classes. We test our approach on the COCO dataset using different splits of strongly and weakly supervised classes. Our approach significantly improves over the Mask R-CNN baseline and obtains competitive performance with the state-of-the-art, while offering a much simpler architecture.