A search is conducted for new phenomena in events with a top quark pair and large missing transverse momentum, where the top quark pair is reconstructed in final states with one isolated electron or muon and multiple jets. The search is performed using the Large Hadron Collider proton-proton collision data sample at a centre-of-mass energy of s = 13 TeV recorded by the ATLAS detector that corresponds to an integrated luminosity of 140 fb−1. An analysis based on neural network classifiers is optimised to search for directly produced pairs of supersymmetric partners of the top quark (stop), and to search for spin-0 mediators, produced in association with a pair of top quarks, that decay into dark-matter particles. In the stop search, the analysis is designed to target models in which the mass difference between the stop and the neutralino from the stop decay is close to the top quark mass. This new search is combined with previously published searches in final states with different lepton multiplicities. No significant excess above the Standard Model background is observed, and limits at 95% confidence level are set. Models with neutralinos with masses up to 570 GeV are excluded, while for small neutralino masses models are excluded for stop masses up to 1230 GeV. Scalar (pseudoscalar) dark matter mediator masses as large as 350 (370) GeV are excluded when the coupling strengths of the mediator to Standard Model and dark-matter particles are both set to one. At lower mediator masses, models with production cross-sections as small as 0.15 (0.16) times the nominal predictions are excluded. Results of this search are also used to set constraints on effective four-fermion contact interactions between top quarks and neutrinos.