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\nContext. Photometric redshifts for galaxies hosting an accreting supermassive black hole in their center, known as active galactic nuclei (AGNs), are notoriously challenging. At present, they are most optimally computed via spectral energy distribution (SED) fittings, assuming that deep photometry for many wavelengths is available. However, for AGNs detected from all-sky surveys, the photometry is limited and provided by a range of instruments and studies. This makes the task of homogenizing the data challenging, presenting a dramatic drawback for the millions of AGNs that wide surveys such as SRG/eROSITA are poised to detect.\nAims. This work aims to compute reliable photometric redshifts for X-ray-detected AGNs using only one dataset that covers a large area: the tenth data release of the Imaging Legacy Survey (LS10) for DESI. LS10 provides deep grizW1-W4 forced photometry within various apertures over the footprint of the eROSITA-DE survey, which avoids issues related to the cross-calibration of surveys.\nMethods. We present the results from C<sc>IRCLE</sc>Z, a machine-learning algorithm based on a fully connected neural network. C<sc>IRCLE</sc>Z is built on a training sample of 14 000 X-ray-detected AGNs and utilizes multi-aperture photometry, mapping the light distribution of the sources.\nResults. The accuracy (σNMAD) and the fraction of outliers (η) reached in a test sample of 2913 AGNs are equal to 0.067 and 11.6%, respectively. The results are comparable to (or even better than) what was previously obtained for the same field, but with much less effort in this instance. We further tested the stability of the results by computing the photometric redshifts for the sources detected in CSC2 and Chandra-COSMOS Legacy, reaching a comparable accuracy as in eFEDS when limiting the magnitude of the counterparts to the depth of LS10.\nConclusions. The method can be applied to fainter samples of AGNs using deeper optical data from future surveys (for example, LSST, Euclid), granting LS10-like information on the light distribution beyond the morphological type. Along with this paper, we have released an updated version of the photometric redshifts (including errors and probability distribution functions) for eROSITA/eFEDS.\n