Accurate segmentation of wound and periwound regions, in clinical images, is critical for effective wound management, which can have a significant impact on patient outcomes and healthcare costs. This paper presents a novel approach to segment wound and periwound regions using weakly supervised learning on partially annotated datasets. We use two public datasets, FUSeg, which provides only wound bed annotations, and WOUNDSEG, which provides the segmentation map for the region containing the periwound and wound bed without explicit distinction. Therefore, we use morphological operations to deal with these incomplete annotations. In addition, we use a pixel-based weighted loss function to improve segmentation accuracy by accounting for the uncertainty inherent in the generated pseudo-labels.