  Recently gate-mediated supercurrent suppression in superconducting
nano-bridges has been reported in many experiments. This could be either a
direct or an indirect gate effect. The microscopic understanding of this
observation is not clear till now. Using the quasiclassical Green's function
method, we show that a small concentration of magnetic impurities at the
surface of the bridges can significantly help to suppress superconductivity and
hence the supercurrent inside the systems while applying a gate field. This is
because the gate field can enhance the depairing through the exchange
interaction between the magnetic impurities at the surface and the
superconductor. We also obtain a \emph{symmetric} suppression of the
supercurrent with respect to the gate field, a signature of a direct gate
effect. Future experiments can verify our predictions by modifying the surface
with magnetic impurities.
