You may be able to just simply drop your bowl and clean things that way. To do this, I work on the quad on the right side (easier access). Remove the side panel by your knees, loosen the boot clamps on both sides of the carb, then rotate the carb so that the bowl (the bottom) is facing you. Remove the 4 screws that hold the bowl onto the carb. When you remove it, gas will come out, so have a rag there to mop it up.
Once off, you'll be able to see what ya got cookin' in there. LIkely some sediment and/or water. You should remove all the jets in there and hit them with carb cleaner spray, then reassemble.
I would also take this time to do two things. First, install a one way check valve onto the drain line coming off the bottom of the bowl. This not only allows fuel to drain if it overflows, but it can allow water UP and into the carb. Installing a one way check valve ($5 or so at your local auto parts store) will cure that. Secondly, I would install a fuel filter between your fuel pump and the carb.
Typically, I recommend taking the carb fully off of the quad and soaking it to clean it. HOwever, in your case, it doesn't sound like you're all varnished up and such, but rather just have junk in your carb bowl that's juking up the works.
Just my thoughts.