  Cosmic Microwave Background (CMB) photons can undergo resonant conversion
into axions in the presence of magnetized plasma distributed inside non-linear
large-scale structure (LSS). This process leads to axion-induced patchy
screening: secondary temperature and polarization anisotropies with a
characteristic non-blackbody frequency dependence that are strongly correlated
with the distribution of LSS along our past light cone. We compute the
axion-induced patchy screening contribution to two- and three- point
correlation functions that include CMB anisotropies and tracers of LSS within
the halo model. We use these results to forecast the sensitivity of existing
and future surveys to photon-axion couplings for axion masses between $2\times
10^{-13}$ eV and $3\times 10^{-12}$ eV, using a combination of empirical
estimates from Planck data of the contribution from instrumental noise and
foregrounds as well as modeled contributions on angular scales only accessible
with future datasets. We demonstrate that an analysis using Planck and the
unWISE galaxy catalogue would be complementary to the most sensitive existing
astrophysical axion searches, probing couplings as small as $3\times 10^{-12}
\, {\rm GeV}^{-1}$, while observations from a future survey such as CMB-S4
could extend this reach by almost an additional order of magnitude.
