  The physics of black holes can suggest new ways to test the existence of
axions. Much work has been done so far to analyse the phenomenon of
superradiance associated with axions in the ergoregion surrounding rotating
black holes. In this work, we instead investigate how Chern-Simons axion
couplings of the form $\phi \,F\,\tilde F$ and $\phi \,R\,\tilde R$, well
motivated by particle physics and string theory, can induce long range profiles
for light axion fields around charged black holes, with or without spin. We
extend known solutions describing axion hairs around spherically symmetric,
asymptotically flat dyonic black hole configurations, charged under $U(1)$
gauge symmetries, by including non-minimal couplings with gravity. The axion
acquires a profile controlled by the black hole conserved charges, and we
analytically determine how it influences the black hole horizon and its
properties. We find a Smarr formula applying to our configurations. We then
generalise known solutions describing axion hairs around slowly rotating black
hole configurations with charge. To make contact with phenomenology, we briefly
study how long range axion profiles induce polarised deflection of light rays,
and the properties of ISCOs for the black hole configurations we investigate.
