  Searches for the exclusive decays of the Higgs boson into $D^*\gamma$ and of
the $Z$ boson into $D^0\gamma$ and $K^0_s\gamma$ can probe flavour-violating
Higgs and $Z$ boson couplings to light quarks. Searches for these decays are
performed with a $pp$ collision data sample corresponding to an integrated
luminosity of $136.3$ fb$^{-1}$ collected at $\sqrt{s}=13$ TeV between
2016-2018 with the ATLAS detector at the CERN Large Hadron Collider. In the
$D^*\gamma$ and $D^0\gamma$ channels, the observed (expected) 95$\%$
confidence-level upper limits on the respective branching fractions are ${\cal
B}(H\rightarrow D^*\gamma)< 1.0 (1.2)\times 10^{-3}$, ${\cal B}(Z\rightarrow
D^0\gamma)< 4.0 (3.4)\times 10^{-6}$, while the corresponding results in the
$K^0_s\gamma$ channel are ${\cal B}(Z\rightarrow K^0_s\gamma)< 3.1 (3.0)\times
10^{-6}$.
