Why does my 2005 360 prairie idle fine when wet but bogs down when u push down on the throttle?

It could be a bad clutch

Check for vacuum leaks. Spay starter fluid around intake,carb base, etc., while engine is idling, listen for change in engine idle.
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Harry, dirty carb or injectors.
Check the attached links,Manuals and guides, Good luck with your Bike
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ATV TROUBLESHOOTING AND MAINTENANCE
ATV Service and Repair Manuals
http://www.carlsalter.com/motorcycle-manuals.asp

'03 400 backfiring/bogging under throttle - ArcticChat.com ...

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Prairie 300 carb problems - YouTube

? 0:28
www.youtube.com/watch?v=w1-znueTqFE
Dec 3, 2012 - Uploaded by Joey Fortner
Prairie 300 carb problems ... The slide (the piece you see moving up and down) has the needle running .... 2005 Kawasaki PrairieIdle issue.

Assuming your diagnosis is good, how about the fuel filter and/or fuel pump?

Spray carb cleaner staight into carb opening at fast idle

If you've rebuilt and adjusted the carbs to the correct spec, and the air fuel mixture is right, it would be worth checking the fuel filter if one is present, as well the fuel pump (again, if present). These are typically the first place to look when you idle fine but die out on throttle.

As for the belt light, it depends on what it indicates. Wear? Tension? If the belt slips, or is too tight or too loose, it could also cause stalling under throttle. With the engine off and cool, check it for cracks, splits and wear, and give it a light tug with two fingers. It should have a small amount of play, but not enough to feel like you could pull it off. If it feels like a steel rod, it may be over tight.

If all of these things are fine, no clogged filters, etc, it's maybe time to look at the distributor, and recheck those carbs.

Camshaft lobes are worn way down, not opening valves enough, and throwing off valve timing. Replace camshaft and followers. Make sure cam chain is still good. If cam chain is good, it's a fairly easy job.