A search for Higgs boson decays into a $Z$ boson and a light resonance in two-lepton plus jet events is performed, using a $pp$ collision dataset with an integrated luminosity of $139\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ collected at $\sqrt{s}=13\text{ }\text{ }\mathrm{TeV}$ by the ATLAS experiment at the CERN LHC. The resonance considered is a light boson with a mass below 4 GeV from a possible extended scalar sector or a charmonium state. Multivariate discriminants are used for the event selection and for evaluating the mass of the light resonance. No excess of events above the expected background is found. Observed (expected) 95% confidence-level upper limits are set on the Higgs boson production cross section times branching fraction to a $Z$ boson and the signal resonance, with values in the range 17--340 pb (${16}_{\ensuremath{-}5}^{+6}--32{0}_{\ensuremath{-}90}^{+130}\text{ }\text{ }\mathrm{pb}$) for the different light spin-0 boson mass and branching fraction hypotheses, and with values of 110 and 100 pb (${100}_{\ensuremath{-}30}^{+40}$ and ${100}_{\ensuremath{-}30}^{+40}\text{ }\text{ }\mathrm{pb}$) for the ${\ensuremath{\eta}}_{c}$ and $J/\ensuremath{\psi}$ hypotheses, respectively.