Tell you what....you'd better give us more info on just what kind of engine/drive train you have. You say it 'takes a bit to kick over'. That almost sounds as if you've got something with a kick starter; like a motorcycle.
But for right now I'll figure that what you've got is a typical chinese 2 stroke with chain drive. And that you're 'roll' starting it.
If that's the case, then it sounds like your clutch is out of adjustment. It's not fully engaging. Maybe it grabs just enough to start but is all but disengaged when the engine spins at higher RPMs.
If you type 'clutch adjustment' in the search box, then you're bound to find all sorts of threads that describe how to get it back in proper adjustment.
Good luck.