We present the combination of the Canada-France-Hawaii Telescope (CHFT) Large Area U-bands Deep Survey (CLAUDS) and the Hyper-Suprime-Cam (HSC) Subaru Strategic Program (HSC-SSP) data over their four deep fields. We provide photometric catalogs for u, u* (CFHT-MegaCam), g, r, i, z, and y (Subaru-HSC) bands over similar to 20 deg(2), complemented in two fields by data from the Visible and Infrared Survey Telescope for Astronomy (VISTA) Deep Extragalactic Observations (VIDEO) survey and the UltraVISTA survey, thus extending the wavelength coverage toward near-infrared with VIRCAM Y, J, H, and K-s observations over 5.5 deg(2). The extraction of the photometry was performed with two different softwares: the HSC pipeline hscPipe and the standard and robust SExtractor software. Photometric redshifts were computed with template-fitting methods using the new Phosphoros code for the hscPipe photometry and the well-known Le Phare code for the SExtractor photometry. The products of these methods were compared with each other in detail. We assessed their quality using the large spectroscopic sample available in those regions, together with photometry and photometric redshifts from COSMOS2020, the latest version of the Cosmic Evolution Survey catalogs. We find that both photometric data sets are in good agreement in Ugrizy down to magnitude similar to 26, and to magnitude similar to 24.5 in the YJHK(s) bands. We achieve good performance for the photometric redshifts, reaching precisions of sigma(NMAD) less than or similar to 0.04 down to m(i) similar to 25, even using only the CLAUDS and HSC bands. At the same magnitude limit, we measured an outlier fraction of eta less than or similar to 10% when using the Ugrizy bands, and down to eta less than or similar to 6% when considering near-infrared data. The hscPipe plus Phosphoros pipeline performs slightly worse in terms of photometric-redshifts precision and outlier fraction than its SExtractor plus Le Phare counterpart, which has essentially been tracked down to differences in the photometry. Thus, this work is also a validation of the Phosphoros code. The photometric catalogs with the data and photometric redshifts from the two pipelines are presented and made publicly available.