  We present a phenomenological study of the single top (anti-)quark production
with leptonic decays at the Large Hadron electron Collider (LHeC) at the
next-to-leading-order (NLO) in QCD. We focus on various differential
distributions in a fiducial region. The NLO corrections can reduce the fiducial
cross section by 14%. We find the NLO predictions exhibit strong stability
under scale variations for most observables considered while the scale
variations at the leading-order (LO) dominated in the theoretical
uncertainties. We propose a method of determining the top-quark mass using the
measurement of the average transverse momentum of the charged lepton. The scale
variations at the NLO induce a theoretical uncertainty of about 1.3 GeV of the
extracted top-quark mass. The statistical error of the extracted top-quark mass
amounts to 1.1 GeV. We also investigate the impact of the QCD corrections and
the scale variations in searches of the anomalous Wtb couplings.
