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.
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