So i have Aprilia Leonardo 125 and i have this weird problem with it. It starts normally and i drive it for as long as i want on half throttle or for a shot period of time on full throttle, but if hold throttle longer and then release, it starts shutting down. When it starts shutting down i can let bike slow down and then try to give some throttle and sometimes it works, but most of the time if i give some throttle it shuts down. After that it starts normally and i can drive it on half throttle without any problems.
Problem I've found with any 2 wheeled vehicle is usually related to the gasoline sitting in the tank, going bad and gumming up the works. I'm taking for granted your Aprllia has a carb or is it fuel injected? If its carbureted, or even fuel injected you might try using a carb cleaner in the gas, or better yet use an Ethenol treatment. Certain times of the year gasoline is blessed with Ethenol for emissions issues. This separates from the gasoline and lines the tank and inside the fuel system. An anti Ethenol treatment like Amsoil Quickshot makes quick work of that. Go by the instructions on the bottle and have at it. They have some at your local parts store as well. I also use "Gumout" fuel injector/carb cleaner. Had people come to me expecting a carb overhaul and two tankfuls with that stuff in the gas and I saved them a couple hundred bucks. Try it, if that don't work,. Then I got more, but you need to start someplace, might as well start out with a cheap fix that has a good track record.
If cleaning the carb doesn't fix the issue, and the engine idles fine, you probably just need to adjust the carburetor mixture. There are plenty of guides online, but once the engine has warmed up, adjust the idle-speed knob (you may need a special tool - Amazon, etc. sell kits) one direction until the engine stutters, and then the other direction until it stutters. You want to leave the knob right in-between those two points.
Once you have the low-speed dialed in, open the throttle to bring the engine up to speed and repeat the adjustment procedure for high speed.