[en] The cross sections for Neutral Current e⁺p Deep Inelastic Scattering (NC DIS) with longitudinally polarised positron beams were measured at a centre-of-mass energy √(s)=318 GeV using the ZEUS detector at HERA. Single-differential cross sections as a function of the virtuality of the exchanged boson, Q², of the inelasticity, y, and of the Bjorken scaling variable, x, and reduced cross sections (as a function of x and Q²) were measured in the phase space region defined as Q²>185 GeV², y<0.9 and y(1-x)²>0.004. The results, based on data corresponding to an integrated luminosity of 135.5 pb^-1, are given for zero polarisation for the whole sample, as well as for both positive and negative values of the longitudinal polarisation of the positron beam. The measured cross sections are compared to the predictions of the Standard Model. The proton structure function xF₃ is measured and the interference term xF^γZ₃ is extracted as a function of x at Q²=1500 GeV² by extrapolating the measurements done at different Q² values. The presented measurements of the polarised e⁺p NC DIS cross sections, the xF₃ structure function and the xF^γZ₃ interference term exceed the precision of previous ZEUS measurements. The uncertainties of the Parton Density Functions (PDFs) for the gluon and the u-valence quark are expected to be reduced at high-x values when the presented results are included in the PDF fits of the proton. A study is presented to further reduce PDF uncertainties for the gluon at very high x (above 10^-1) through the inclusion of DIS dijet cross sections in a ZEUS-JETS-like PDF fit. An irradiation facility to study the X-ray-induced radiation damage of silicon sensors for the European XFEL was set up at HASYLAB. Nine gate-controlled diodes were irradiated with doses from 1 kGy to 1 GGy in several irradiation steps and the surface current I_surf and the flatband voltage V_fb were measured after each irradiation. It was found that both, I_surf and V_fb, strongly increase at low doses, but decrease at high doses (above 5 MeV). No evidence was found that this decrease was caused by temperature-induced annealing.