  A search for the decay of the Higgs boson to a $Z$ boson and a light,
pseudoscalar particle, $a$, decaying respectively to two leptons and to two
photons is reported. The search uses the full LHC Run 2 proton-proton collision
data at $\sqrt{s}=13$ TeV, corresponding to 139 fb$^{-1}$ collected by the
ATLAS detector. This is one of the first searches for this specific decay mode
of the Higgs boson, and it probes unexplored parameter space in models with
axion-like particles (ALPs) and extended scalar sectors. The mass of the $a$
particle is assumed to be in the range 0.1-33 GeV. The data are analysed in two
categories: a merged category where the photons from the $a$ decay are
reconstructed in the ATLAS calorimeter as a single cluster, and a resolved
category in which two separate photons are detected. The main background
processes are from Standard Model $Z$ boson production in association with
photons or jets. The data are in agreement with the background predictions, and
upper limits on the branching ratio of the Higgs boson decay to $Za$ times the
branching ratio $a\to\gamma\gamma$ are derived at the 95% confidence level and
they range from 0.08% to 2% depending on the mass of the $a$ particle. The
results are also interpreted in the context of ALP models.
