Cross-section measurements of top-quark pair production where the hadronically decaying top quark has transverse momentum greater than 355 GeV and the other top quark decays into l nu b are presented using 139 fb(-1) of data collected by the ATLAS experiment during proton-proton collisions at the LHC. The fiducial cross-section at root s = 13 TeV is measured to be sigma = 1.267 +/- 0.005 +/- 0.053 pb, where the uncertainties reflect the limited number of data events and the systematic uncertainties, giving a total uncertainty of 4.2%. The cross-section is measured differentially as a function of variables characterising the t (t) over bar system and additional radiation in the events. The results are compared with various Monte Carlo generators, including comparisons where the generators are reweighted to match a parton-level calculation at next-to-next-to-leading order. The reweighting improves the agreement between data and theory. The measured distribution of the top-quark transverse momentum is used to search for new physics in the context of the effective field theory framework. No significant deviation from the Standard Model is observed and limits are set on the Wilson coefficients of the dimension-six operators O-tG and O-tq((8)), where the limits on the latter are the most stringent to date.