A search for decays of the Higgs boson into a $Z$ boson and a light resonance, with a mass of 0.5-3.5 GeV, is performed using the full 140 fb$^{-1}$ dataset of 13 TeV proton-proton collisions recorded by the ATLAS detector during Run 2 of the LHC. Leptonic decays of the $Z$ boson and hadronic decays of the light resonance are considered. The resonance can be interpreted as a $J/ψ$ or $η_c$ meson, an axion-like particle, or a light pseudoscalar in two-Higgs-doublet models. Due to its low mass, it would be produced with high boost and reconstructed as a single small-radius jet of hadrons. A neural network is used to correct the Monte Carlo simulation of the background in a data-driven way. Two additional neural networks are used to distinguish signal from background. A binned profile-likelihood fit is performed on the final-state invariant mass distribution. No significant excess of events relative to the expected background is observed, and upper limits at 95% confidence level are set on the Higgs boson's branching fraction to a $Z$ boson and a light resonance. The exclusion limit is ~10% for the lower masses, and increases for higher masses. Upper limits on the effective coupling $C^\text{eff}_{ZH}/Λ$ of an axion-like particle to a Higgs boson and $Z$ boson are also set at 95% confidence level, and range from 0.9 to 2 TeV$^{-1}$.