A search for new phenomena with top quark pairs in final states with one isolated electron or muon, multiple jets, and large missing transverse momentum is performed. Signal regions are designed to search for two-, three-, and four-body decays of the directly pair-produced supersymmetric partner of the top quark (stop). Additional signal regions are designed specifically to search for spin-0 mediators that are produced in association with a pair of top quarks and decay into a pair of dark-matter particles. The search is performed using the Large Hadron Collider proton-proton collision dataset at a centre-of-mass energy of s = 13 TeV recorded by the ATLAS detector from 2015 to 2018, corresponding to an integrated luminosity of 139 fb⁻¹. No significant excess above the Standard Model background is observed, and limits at 95% confidence level are set in the stop-neutralino mass plane and as a function of the mediator mass or the dark-matter particle mass. Stops are excluded up to 1200 GeV (710 GeV) in the two-body (three-body) decay scenario. In the four-body scenario stops up to 640 GeV are excluded for a stop-neutralino mass difference of 60 GeV. Scalar and pseudoscalar dark-matter mediators are excluded up to 200 GeV when the coupling strengths of the mediator to Standard Model and dark-matter particles are both equal to one and when the mass of the dark-matter particle is 1 GeV. [Figure not available: see fulltext.]