Could be a handful of things; 2 that you could check and let me know what the outcome is will narrow it down -
1. When was the last clutch replacement done on the bike if any? When your clutch starts to go - one of the symptoms is loss of power at take off.
Here's something you can do and I believe the 350 had a basket as well as a centrifugal clutch if you have an adjustment - there should be a nut on a stud on the front cover of the engine. Front Housing is what it's called. You can't miss the stud - it's on the engine cover that faces forward. That's your clutch adjustment. Hold the stud stationary and loosen the nut - should be a 10 or 12mm - once the nut is backed off - loosen the stud - then go counter clockwise slowly until you start to feel resistance - once you feel resistance - go forward about a 1/4 turn.
That's a standard clutch adjustment for multiple makes & models especially the Foreman, etc. I've got a 2000 TRX450 and that's the clutch adjustment*. If this atv has the hondamatic - or no clutch adjustment stud/nut - I'd pull that front cover off and check the tolerance on your centrifugal clutch.
2. - every 6 months or 500 hours they recommend you check and adjust your valves as needed. No one does it and the valves get out of whack. What could possibly be happening - your valves need adjustment - when you start the bike and until it's really warmed up performance will suffer. Once they bike warms up, heat makes everything expand - and if you're on that threshold - you'll run terrible at first/better as the valves warm up. You'll see this play out everytime you get on it and it hasn't been running in 20 hours.
Could be a carb issue as well, your idle and slow jets could be getting clogged up. Or your pilot screw's o-ring is shot and you're drawing too much air at idle/slow speeds and evens out under demand. Once that happens - you'll probably idle okay/nothing great - putting around will be about the same - but when you're on full throttle your carb pulls from a different jet. Each jet has a job - idle jet, slow jet, fast jet, - if you haven't cleaned the inside of that carb out in awhile - now is the time to do it - if nothing else, you'll have a clean carb and you'll have that knocked off your list.
These aren't the only causes by far - there's a flow chart attached/it's from an 04 TRX400 - Same rules apply. I didn't touch on electrical - but if you check on the things above/need any tips going through the flow chart, etc. feel free to shoot me a message. Glad to help if possible.
All The Best, Justin Lee, 17 South Industries, LLC