  (abridged)Photometric redshifts for AGN (galaxies hosting an accreting
supermassive black hole in their center) are notoriously challenging and
currently better computed via SED fitting, assuming that deep photometry for
many wavelengths is available. However, for AGN detected all-sky, the
photometry is limited and provided by different projects. This makes the task
of homogenising the data challenging and is a dramatic drawback for the
millions of AGN that wide surveys like SRG/eROSITA will detect. This work aims
to compute reliable photometric redshifts for X-ray-detected AGN using only one
dataset that covers a large area: the 10th Data Release of the Imaging Legacy
Survey (LS10) for DESI. LS10 provides deep grizW1-W4 forced photometry within
various apertures, thus avoids issues related to the cross-calibration of
surveys. We present the results from CircleZ, a machine-learning algorithm
based on a Fully Connected Neural Network. CircleZ uses training sample of
14,000 X-ray-detected AGN and utilizes multi-aperture photometry. The accuracy
and the fraction of outliers reached in a test sample of 2913 AGN are 0.067 and
11.6%, respectively. The results are comparable to or better than those
obtained previously for the same field but with much less effort. We further
tested the stability of the results by computing the photometric redshifts for
the sources detected in CSC2 and Chandra-COSMOS Legacy, reaching comparable
accuracy as in eFEDS when limiting the magnitude of the counterparts with
respect to the depth of LS10. The method applies to fainter samples of AGN
using deeper optical data from future surveys (e.g., LSST, Euclid), granted
LS10-like information on the light distribution beyond a morphological type is
provided. With the paper, we release an updated version of the photometric
redshifts (including errors and probability distribution function) for
eROSITA/eFEDS.
