  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 $\sim 20~{\rm deg}^2$, 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 $5.5~{\rm deg}^2$. 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$\sim26$, and to magnitude$\sim24.5$ in the $YJHK_s$ bands. We achieve
good performance for the photometric redshifts, reaching precisions of
$\sigma_{NMAD} \lesssim 0.04$ down to ${m}_i\sim25$, even using only the CLAUDS
and HSC bands. At the same magnitude limit, we measured an outlier fraction of
$\eta \lesssim 10\%$ when using the $Ugrizy$ bands, and down to $\eta \lesssim
6\%$ when considering near-infrared data. [abridged]
