A search for a new heavy boson produced via gluon-fusion in the four-lepton channel with missing transverse momentum or jets is performed. The search uses proton-proton collision data equivalent to an integrated luminosity of 139 fb$^{−1}$ at a centre-of-mass energy of 13 TeV collected by the ATLAS detector between 2015 and 2018 at the Large Hadron Collider. This study explores the decays of heavy bosons: R → SH and A → ZH, where R is a CP-even boson, A is a CP-odd boson, H is a CP-even boson, and S is considered to decay into invisible particles that are candidates for dark matter. In these processes, S → invisible and H → ZZ. The Z boson associated with the heavy scalar boson H decays into all decay channels of the Z boson. The mass range under consideration is 390–1300 (320–1300) GeV for the R (A) boson and 220–1000 GeV for the H boson. No significant deviation from the Standard Model backgrounds is observed. The results are interpreted as upper limits at a 95% confidence level on the cross-section times the branching ratio of the heavy resonances.[graphic not available: see fulltext]