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A search for Higgs boson decays into a $Z$ boson and a light resonance intwo-lepton plus jet events is performed, using a $pp$ collision dataset with anintegrated luminosity of 139 fb$^{-1}$ collected at $\sqrt{s}=13$ TeV by theATLAS experiment at the CERN LHC. The resonance considered is a light bosonwith a mass below 4 GeV from a possible extended scalar sector, or a charmoniumstate. Multivariate discriminants are used for the event selection and forevaluating the mass of the light resonance. No excess of events above theexpected background is found. Observed (expected) 95$\% $ confidence-levelupper limits are set on the Higgs boson production cross section timesbranching fraction to a $Z$ boson and the signal resonance, with values in therange 17 pb to 340 pb ($16^{+6}_{-5}$ pb to $320^{+130}_{-90}$ pb) for thedifferent light spin-0 boson mass and branching fraction hypotheses, and withvalues of 110 pb and 100 pb ($100^{+40}_{-30}$ pb and $100^{+40}_{-30}$ pb) forthe $\eta_c$ and $J/\psi$ hypotheses, respectively.

Search for Higgs boson decays into a $Z$ boson and a light hadronically decaying resonance using 13 TeV $pp$ collision data from the ATLAS detector