The abundance of matter to antimatter in the universe is not explained by the Standard Model (SM). Two Higgs Doublet Models (2HDMs), which introduce additional Higgs particles, and in particular the A→ZH channel may partly explain the origin of the matter asymmetry. A search for the A→ZH signature is done using 139 fb⁻¹ of integrated luminosity at a center-of-mass energy of √s=13 TeV recorded by ATLAS during 2015–2018 proton–proton collisions at the LHC. The heavy Aboson (300<m_A<800 GeV) is produced by gluon–gluon fusion. The heavy H boson (200<m_H<700 GeV) decays to WW that further decay to qqqq. Z decays to a pair of electrons or muons. The individual W bosons are reconstructed from the final state quarks. Signal shapes are interpolated from Monte Carlo simulations. The rates of known (background) processes are constrained by dedicated control regions in data. No excess above the SM predictions is found. The upper limits at the 95% confidence level for σ(gg→A)×BR(A→ZH)×BR(H→WW) are set to 0.023–8.9 pb, depending on mass. Using the results and taking the natural width of A into account, exclusions are placed on the phase space of type-I 2HDMs. In a separate performance analysis, the electron identification of the ATLAS Transition Radiation Tracker is calibrated using Z→ee, μμ events obtained by the tag and probe method applied to LHC data recorded by ATLAS in 2016 as well as simulated data. The new calibration solves the issue of a rising number of electrons with low probabilities. The background efficiency is reduced by 1.45 (3.31) percentage points in data (simulation) at 95% signal efficiency compared to the previous calibration.