  Searches for the exclusive decays of the Higgs boson to an $\omega$ meson and
a photon or a $K^{*}$ meson and a photon can probe flavour-conserving and
flavour-violating Higgs boson couplings to light quarks, respectively. Searches
for these decays, along with the analogous $Z$ boson decay to an $\omega$ meson
and a photon, are performed with a $pp$ collision data sample corresponding to
integrated luminosities of up to 134 fb$^{-1}$ collected at $\sqrt{s}=13$ TeV
with the ATLAS detector at the CERN Large Hadron Collider. The obtained 95%
confidence-level upper limits on the respective branching fractions are ${\cal
B}(H\rightarrow\omega\gamma)< 5.5\times 10^{-4}$, ${\cal B}(H\rightarrow
K^{*}\gamma)< 2.2\times10^{-4}$ and ${\cal B}(Z\rightarrow
\omega\gamma)<3.9\times 10^{-6}$. The limits for $H\rightarrow \omega\gamma$
and $Z\rightarrow \omega\gamma$ are 370 times and 140 times the Standard Model
expected values, respectively. The result for $Z\rightarrow \omega\gamma$
corresponds to a two-orders-of-magnitude improvement over the limit obtained by
the DELPHI experiment at LEP.
