What can I checkdo to stop my son's 05 ttr90e spitting and sputtering?

Ours was doing the same thing. Open the drain at the bottom of the bowl with a clear container under the small hose and see what comes out. If it is nasty, you need to do some cleaning. Remove the carburator, take it apart, clean the bowl and all the jets. I use contact cleaner after removing the rubber bowl gasket and blow it thru the jets and orifices after removing the jets. Really a simple carb if you are at all mechanically inclined. These carbs seem to easily get gunked up.................