We have measured IV-characteristics of a superconducting single-electron transistor (SET) whose island consists of SNS (Al-Cu-Al or Al-Cr-Al) structure with lengths 0.4, 0.15 and 0.4μm, respectively. In samples with copper, a supercurrent as well as resonances are observed at small bias, and the samples behave like regular SSS SETs. The amplitude of Josephson-quasiparticle (JQP) peaks, however, displays strong variation with gate modulation, even though the Coulomb energy is small (island capacitance ~1 fF). Chrome sections on the island result in a smooth, exponentially varying sub-gap current that can be modeled with the voltage fluctuation theory.