The study of a Kondo impurity in BCS superconductors stretches back several decades. It has been found by several authors that the impurity is screened by a local bound state – the Yu-Shiba-Rusinov (YSR) intra-gap states. Here we study a system that consists of a one-dimensional charge conserving spin-singlet superconductor coupled to a magnetic impurity at the boundary. In such system quantum fluctuations are enhanced and we show that they lead to the destruction of Yu-Shiba-Rusinov (YSR) intra-gap bound-states in all but a narrow region of the phase diagram, as opposed to a Kondo impurity in a BCS superconductor where the Shiba state occurs over the whole phase. We find that for large enough impurity spin exchange interactions a renormalized Kondo-screened regime is established. In this regime, not found for BCS superconductors, there are no YSR states and a renormalized Kondo scale is generated. Another regime, not present in BCS superconductors, is the moment regime that occurs for weak coupling. We also study the system with two Kondo impurities attached to the wire, one at each edge and find a boundary SUSY emerging. It reduces to boundary Majorana modes when a strong edge magnetic field is applied.