My Kymco 250 made a whining noice when it started and I had to replace the belt at 12,000 km. It had started squealing every now and then when I applied throttle to accelerate from a stop. It didn’t make the noise when starting the engine though.
The issue was due to the belt stretching. The belt width was still fine. Someone else reported the same problem at about the same time. You could check by removing the belt cover – the silver painted cover below the black air cleaner case. If you can pinch the top & bottom of the belt together in the middle (or come close), it has stretched.
If the issue you are having is due to the belt it is fairly easy to check.