The recent growth of large foundation models that can easily generate pseudo-labels for huge quantity of unlabeled data makes unsupervised Zero-Shot Cross-Domain Image Retrieval (UZS-CDIR) less relevant. In this paper, we therefore turn our attention to weakly supervised ZS-CDIR (WSZS-CDIR) with noisy pseudo labels generated by large foundation models such as CLIP. To this end, we propose CLAIR to refine the noisy pseudo-labels with a confidence score from the similarity between the CLIP text and image features. Furthermore, we design inter-instance and inter-cluster contrastive losses to encode images into a class-aware latent space, and an inter-domain contrastive loss to alleviate domain discrepancies. We also learn a novel cross-domain mapping function in closed-form, using only CLIP text embeddings to project image features from one domain to another, thereby further aligning the image features for retrieval. Finally, we enhance the zero-shot generalization ability of our CLAIR to handle novel categories by introducing an extra set of learnable prompts. Extensive experiments are carried out using TUBerlin, Sketchy, Quickdraw, and DomainNet zero-shot datasets, where our CLAIR consistently shows superior performance compared to existing state-of-the-art methods.