Methods to detect malignant lesions from screening mammograms are usually trained with fully annotated datasets, where images are labelled with the localisation and classification of cancerous lesions. However, real-world screening mammogram datasets commonly have a subset that is fully annotated and another subset that is weakly annotated with just the global classification (i.e., without lesion localisation). Given the large size of such datasets, researchers usually face a dilemma with the weakly annotated subset: to not use it or to fully annotate it. The first option will reduce detection accuracy because it does not use the whole dataset, and the second option is too expensive given that the annotation needs to be done by expert radiologists. In this paper, we propose a middle-ground solution for the dilemma, which is to formulate the training as a weakly- and semi-supervised learning problem that we refer to as malignant breast lesion detection with incomplete annotations. To address this problem, our new method comprises two stages, namely: (1) pre-training a multi-view mammogram classifier with weak supervision from the whole dataset, and (2) extending the trained classifier to become a multi-view detector that is trained with semi-supervised student-teacher learning, where the training set contains fully and weakly-annotated mammograms. We provide extensive detection results on two real-world screening mammogram datasets containing incomplete annotations and show that our proposed approach achieves state-of-the-art results in the detection of malignant breast lesions with incomplete annotations.