A search for the production of a Higgs boson in association with a single top quark, tH, is presented. The analysis uses proton-proton collision data corresponding to an integrated luminosity of 140 fb$^{−1}$ at a centre-of-mass energy of 13 TeV, collected by the ATLAS detector at the LHC. The search targets Higgs-boson decays into $ b\overline{b} $, WW$^{*}$, ZZ$^{*}$, and ττ, accompanied by an isolated lepton (electron or muon) from the top-quark decay. Multivariate techniques are employed to enhance the separation between signal and background processes. The observed signal strength, μ$_{tH}$, defined as the ratio between the measured cross-section and the predicted Standard Model value, is μ$_{tH}$ = 8.1 ± 2.6 (stat.) ± 2.0 (syst.). The significance of the observed (expected) signal above the background-only expectation is 2.8 (0.4) standard deviations. The corresponding observed (expected) upper limit at the 95% confidence level on the tH cross-section is found to be 13.9 (6.1) times the value predicted by the Standard Model. An interpretation with an inverted sign of the top-quark Yukawa coupling is performed, and the signal strength and corresponding limit are reported.