  Axion couplings to photons could induce photon-axion conversion in the
presence of magnetic fields in the Universe. This conversion could impact
various cosmic distance measurements, such as luminosity distances to type Ia
supernovae and angular distances to galaxy clusters, in different ways. In this
paper we consider different combinations of the most up-to-date distance
measurements to constrain the axion-photon coupling. Employing the conservative
cell magnetic field model for the magnetic fields in the intergalactic medium
(IGM) and ignoring the conversion in the intracluster medium (ICM), we find the
upper bounds on axion-photon couplings to be around $5 \times 10^{-12}$
(nG/$B$) $\sqrt{\mathrm{Mpc/s}}$ GeV$^{-1}$ for axion masses $m_a$ below
$10^{-13}$ eV, where $B$ is the strength of the IGM magnetic field, and $s$ is
the comoving size of the magnetic domains. When including the conversion in the
ICM, the upper bound is lowered and could reach $5 \times 10^{-13}\,
$GeV$^{-1}$ for $m_a < 5 \times 10^{-12}$ eV. While this stronger bound depends
on the ICM modeling, it is independent of the strength of the IGM magnetic
field, for which there is no direct evidence yet. These constraints could be
placed on firmer footing with an enhanced understanding and control of the
astrophysical uncertainties associated with the IGM and ICM. All the bounds are
determined by the shape of the Hubble rate as a function of redshift
reconstructable from various distance measurements, and insensitive to today's
Hubble rate, of which there is a tension between early and late cosmological
measurements. As an appendix, we discuss the model building challenges of the
use of photon-axion conversion to make type Ia supernovae brighter to alleviate
the Hubble problem/crisis.
