We excite exciton-polariton condensates in half-moon shapes by the non-resonant opti cal excitation of GaAs-based cylindrical pillar microcavities. In this geometry, the 1r-jump of the phase of the condensate wave function coexists with a gradual ±1r phase variation between two horns of the half-moon. We switch between clockwise and counter-clockwise phase currents by slightly shifting the excitation spot on the surface of the pillar. Half moon condensates are expected to reveal features of two-level quantum systems similar to superconducting flux qubits.Among the most well studied states with non-zero OAM are vortex states characterized by the azimuthal phase dependence of the form exp(im0), where the integer mis a quantum num ber frequently referred to as a topological charge [4,5], and nm is the OAM per photon [6], 0 is the azimuthal angle. There exist several approaches for generation of states with non-integer orbital momenta based on the introduction of local phase jumps that make the phase varia tion in the remaining part of the closed loop different from a multiple of 21r. Depending on the physical system under consideration, various approaches are used to create quantum states with the non-integer OAM. For generation of optical fractional vortices one needs specially designed optical elements, fractional vortex lenses or spiral phase plates [7][8][9]. In superfluid ring-shaped Bose-Einstein condensates of ultracold atoms, the method in demand for generating flow states characterized by a phase slip consists in the insertion of a so-called weak link [10,11]. A weak link represents a depletion in the condensate density produced by the external potential barrier moving along the circumference of the condensate at a given velocity. It induces back-scattering for the condensate flows spreading clockwise and counter-clockwise along the ring. This results in interference patterns of the scattered flow states. In recent years, a significant progress in shaping of Bose-Einstein condensates of light-matter quasiparticles, exciton-polaritons, has been reported [12,13]. In contrast to atomic condensates, polariton condensates may be gener ated at relatively high temperatures, including the room temperature [14] that opens the way to their application in classical and quantum information processing.In the present work, creation of polariton phase currents characterised by fractional topolog ical charges is demonstrated by the near-field optical interferometry in cylindrical micropillar microcavities. We experimentally study semiconductor micropillars where the demonstration of concentric circular and lobe condensate patterns [15] and generation of persistent circular currents [16] were reported earlier. We introduce a weak link in a circular polariton condensate that takes a half-moon shape. The polariton density depletion between the horns of the half moon is provoked by the repulsive interaction of the condensate polaritons with the reservoir of incoherent excitons generated by a non-resonant optical excitation und...