Measurements of the cross sections for charged current deep inelastic scattering in e⁺p collisions with a longitudinally polarised positron beam are presented. The measurements are based on a data sample with an integrated luminosity of 132 pb^-1 collected with the ZEUS detector at HERA at a centre-of-mass energy of 318 GeV. The total cross section is presented at positive and negative values of the longitudinal polarisation of the positron beams. The single-differential cross-sections dσ/dQ², dσ/dx and dσ/dy are presented for Q²>200 GeV². The reduced cross-section σ̃ is presented in the kinematic range 200<Q²<60000 GeV² and 0.006<x<0.562. The measurements agree well with the predictions of the Standard Model. The results are used to determine a lower limit on the mass of a hypothetical right-handed W boson.