No. The MiniProg3 cannot be used to debug PSoC1 devices. PSoC3 and PSoC5 parts have on-chip JTAG and SWD debug features which are used by the MiniProg3 to debug them.
But PSoC1 parts do not have these debugging features. So, to debug PSoC1 project ICE and POD are required. MiniProg3 can not be used to debug PSoC1 based applications