Training a model to perform a task typically requires a large amount of data from the domains in which the task will be applied. However, it is often the case that data are abundant in some domains but scarce in others. Domain adaptation deals with the challenge of adapting a model trained from a data-rich source domain to perform well in a data-poor target domain. In general, this requires learning plausible mappings between domains. CycleGAN is a powerful framework that efficiently learns to map inputs from one domain to another using adversarial training and a cycle-consistency constraint. However, the conventional approach of enforcing cycle-consistency via reconstruction may be overly restrictive in cases where one or more domains have limited training data. In this paper, we propose an augmented cyclic adversarial learning model that enforces the cycle-consistency constraint through an external task specific model, which encourages the preservation of task-relevant content as opposed to exact reconstruction. We explore digit classification with MNIST and SVHN in a low-resource setting in supervised, semi and unsupervised situation. In low-resource supervised setting, the results show that our approach improves absolute performance by $14\%$ and $4\%$ when adapting SVHN to MNIST and vice versa, respectively, which outperforms unsupervised domain adaptation methods that require high-resource unlabeled target domain. Moreover, using only few unsupervised target data, our approach can still outperforms many high-resource unsupervised models. In speech domains, we also adopt a speech recognition model from each domain as the task specific model. Our approach improves absolute performance of speech recognition by $2\%$ for female speakers in the TIMIT dataset, where the majority of training samples are from male voices.