High precision single-differential W +/--boson production cross-sections as a function of electron or muon transverse momentum pT or their pseudorapity., as well as double-differential cross-sections as functions of these variables, are measured in proton-proton collisions at centre-of-mass energies root s = 5.02TeV and 13TeV. The W-boson charge asymmetry as a function of lepton. is also measured. The data, collected in dedicated runs at reduced instantaneous luminosity with the ATLAS detector at the Large Hadron Collider, correspond to integrated luminosities of 255 pb(-1) at 5.02TeV and 338 pb(-1) at 13TeV. The measurements are in agreement with Standard-Model predictions calculated at next-to-next-to-leading-order in the strong coupling constant as including transverse-momentum resummation at next-to-next-to-leading logarithmic accuracy using several parton distribution functions. The impact of the measured differential cross-sections as a function of lepton. on the determination of these functions is studied using a profiling technique.