Axion-like particles (ALPs) are well-motivated extensions of the Standard Model of Particle Physics and a generic prediction of some string theories. X-ray observations of bright active galactic nuclei (AGNs) hosted by rich clusters of galaxies are excellent probes of very-light ALPs, with masses
log
(
m
a
/
eV
)
<
−
12.0
. We evaluate the potential of future X-ray observatories, particularly Athena and the proposed AXIS, to constrain ALPs via observations of cluster-hosted AGNs, taking NGC 1275 in the Perseus cluster as our exemplar. Assuming perfect knowledge of the instrument calibration, we show that a modest exposure (200 ks) of NGC 1275 by Athena permits us to exclude all photon–ALP couplings g
aγ
> 6.3 × 10−14 GeV−1 at the 95% confidence level, as previously shown by Conlon et al., representing a factor of 10 improvement over current limits. We then proceed to assess the impact of realistic calibration uncertainties on the Athena projection by applying a standard Cash likelihood procedure, showing the projected constraints on g
aγ
weaken by a factor of 10 (back to the current most sensitive constraints). However, we show how the use of a deep neural network can disentangle the energy-dependent features induced by instrumental miscalibration and those induced by photon–ALP mixing, allowing us to recover most of the sensitivity to the ALP physics. In our explicit demonstration, the machine learning applied allows us to exclude g
aγ
> 2.0 × 10−13 GeV−1, complementing the projected constraints of next-generation ALP dark matter birefringent cavity searches for very-light ALPs. Finally, we show that a 200 ks AXIS/on-axis observation of NGC 1275 will tighten the current best constraints on very-light ALPs by a factor of 3.