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Unsupervised domain adaptation (UDA) aims to transfer knowledge learned from a labeled source domain to an unlabeled target domain. Contrastive learning (CL) in the context of UDA can help to better separate classes in feature space. However, in image segmentation, the large memory footprint due to the computation of the pixel-wise contrastive loss makes it prohibitive to use. Furthermore, labeled target data is not easily available in medical imaging, and obtaining new samples is not economical. As a result, in this work, we tackle a more challenging UDA task when there are only a few (fewshot) or a single (oneshot) image available from the target domain. We apply a style transfer module to mitigate the scarcity of target samples. Then, to align the source and target features and tackle the memory issue of the traditional contrastive loss, we propose the centroid-based contrastive learning (CCL) and a centroid norm regularizer (CNR) to optimize the contrastive pairs in both direction and magnitude. In addition, we propose multi-partition centroid contrastive learning (MPCCL) to further reduce the variance in the target features. Fewshot evaluation on MS-CMRSeg dataset demonstrates that ConFUDA improves the segmentation performance by 0.34 of the Dice score on the target domain compared with the baseline, and 0.31 Dice score improvement in a more rigorous oneshot setting.