Measurement of Charged Current Deep Inelastic Scattering Cross Sections with a Longitudinally Polarised Positron Beam at HERA ZEUS Collaboration 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 135.5 pb^-1 collected with the ZEUS detector at HERA in 2006 and 2007 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 dsigma/dQ^2, dsigma/dx and dsigma/dy are presented for Q^2>200 GeV. The double-differential cross-section d^2sigma/dxdQ^2 is presented in the kinematic range 280