We realize growth of self-catalyzed core-shell GaAs/GaAsP nanowires (NWs) on Si substrates using molecular-beam epitaxy. Transmission electron microscopy (TEM) of single GaAs/GaAsP NWs confirms their high crystal quality and shows domination of the zinc-blende phase. This is further confirmed in optics of single NWs, studied using cw and time-resolved photoluminescence (PL). A detailed comparison with uncapped GaAs NWs emphasizes the effect of the GaAsP capping in suppressing the non-radiative surface states: significant PL enhancement in the core-shell structures exceeding 2000 times at 10K is observed; in uncapped NWs PL is quenched at 60K whereas single core-shell GaAs/GaAsP NWs exhibit bright emission even at room temperature. From analysis of the PL temperature dependence in both types of NW we are able to determine the main carrier escape mechanisms leading to the PL quench.