  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 (AGN) hosted by rich
clusters of galaxies are excellent probes of very-light ALPs, with masses
$\mathrm{log}(m_\mathrm{a}/\mathrm{eV}) < -12.0$. We evaluate the potential of
future X-ray observatories, particularly $Athena$ and the proposed $AXIS$, to
constrain ALPs via observations of cluster-hosted AGN, taking NGC 1275 in the
Perseus cluster as our exemplar. Assuming perfect knowledge of instrument
calibration, we show that a modest exposure (200-ks) of NGC 1275 by $Athena$
permits us to exclude all photon-ALP couplings $g_\mathrm{a\gamma} > 6.3 \times
10^{-14} \ {\mathrm{GeV}}^{-1}$ at the 95% level, as previously shown by
$Conlon \ et \ al. \ (2018)$, 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 $g_\mathrm{a\gamma}$
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 $g_\mathrm{a\gamma} > 2.0 \times 10^{-13} \
{\mathrm{GeV}}^{-1}$, 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.
