  We study the performance of the hybrid template-machine-learning photometric
redshift (photo-$z$) algorithm Delight, which uses Gaussian processes, on a
subset of the early data release of the Physics of the Accelerating Universe
Survey (PAUS). We calibrate the fluxes of the $40$ PAUS narrow bands with $6$
broadband fluxes ($uBVriz$) in the COSMOS field using three different methods,
including a new method which utilises the correlation between the apparent size
and overall flux of the galaxy. We use a rich set of empirically derived galaxy
spectral templates as guides to train the Gaussian process, and we show that
our results are competitive with other standard photometric redshift
algorithms. Delight achieves a photo-$z$ $68$th percentile error of
$\sigma_{68}=0.0081(1+z)$ without any quality cut for galaxies with
$i_\mathrm{auto}<22.5$ as compared to $0.0089(1+z)$ and $0.0202(1+z)$ for the
BPz and ANNz2 codes, respectively. Delight is also shown to produce more
accurate probability distribution functions for individual redshift estimates
than BPz and ANNz2. Common photo-$z$ outliers of Delight and BCNz2 (previously
applied to PAUS) are found to be primarily caused by outliers in the narrowband
fluxes, with a small number of cases potentially indicating spectroscopic
redshift failures in the reference sample. In the process, we introduce
performance metrics derived from the results of BCNz2 and Delight, allowing us
to achieve a photo-$z$ quality of $\sigma_{68}<0.0035(1+z)$ at a magnitude of
$i_\mathrm{auto}<22.5$ while keeping $50$ per cent objects of the galaxy
sample.
