yes turbos will definitely increase engine performance on a small engine. You're essentially increasing the displacement of the hawk, which would drastically increase performance, even if you were to do a moderate blow through.
The stock hawk has a pretty low compression ratio already, supposedly 9:1, however I have a large suspect that it is actually even lower. the tt250(which is the same engine) has a compression ratio of 8.7:1, while at the same time having a thinner head gasket. so i think the hawk has an even lower ratio than that.
You would need to buy the world's smallest turbo, the RHB31. this website sells clones of it for not too expensive of a price.
http://www.ecotrons.com/products/sma..._turbocharger/
You can turbo a carburetor, and there are tuning ways to do so, however it's likely you'll run into fuel issues, as it needs different amounts of fuel depending on how fast the turbo is spinning. Id definitely go fuel injection, they have a lot, although it is expensive, it'll probably save ypu from blowing up an engine
But to be fair, one large issue you may run into is cooling. turbos produce alot of heat, and an intercooler of some kind may be necessary to keep combustion temperatures down. Id definitely run as rich as you can, especially at WOT.
I've thought alot about doing a turbo on my hawk. i decided a supercharger would be better for an off-road purpose bike(less intense power bands generally, don't have to get it spooled up), but the biggest thing stopping me is the price.
If you were set on doing carbs and turbo, I say blow through for better throttle response