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Deep-learning algorithms for cell segmentation typically require large data sets with high-quality annotations to be trained with. However, the annotation cost for obtaining such sets may prove to be prohibitively expensive. Our work aims to reduce the time necessary to create high-quality annotations of cell images by using a relatively small well-annotated data set for training a convolutional neural network to upgrade lower-quality annotations, produced at lower annotation costs. We investigate the performance of our solution when upgrading the annotation quality for labels affected by three types of annotation error: omission, inclusion, and bias. We observe that our method can upgrade annotations affected by high error levels from 0.3 to 0.9 Dice similarity with the ground-truth annotations. We also show that a relatively small well-annotated set enlarged with samples with upgraded annotations can be used to train better-performing cell segmentation networks compared to training only on the well-annotated set. Moreover, we present a use case where our solution can be successfully employed to increase the quality of the predictions of a segmentation network trained on just 10 annotated samples.