  It is extremely challenging to probe the charm-quark Yukawa coupling at
hadron colliders primarily due to the large Standard Model (SM) background
(including $h\to b\bar b$) and the lack of an effective trigger for the signal
$h\to c\bar c$. We examine the feasibility of probing this coupling at the LHC
via a Higgs radiative decay $h\rightarrow c\bar{c}\gamma$. The existence of an
additional photon in the final state may help for the signal identification and
background suppression. Adopting a refined triggering strategy and utilizing
basic machine learning, we find that a coupling limit of about 8 times the SM
value may be reached with $2\sigma$ sensitivity after the High Luminosity LHC
(HL-LHC). Our result is comparable and complementary to other projections for
direct and indirect probes of $h\to c\bar c$ at the HL-LHC. Without a
significant change in detector capabilities, there would be no significant
improvement for this search from higher energy hadron colliders.
