There is a diapragm on the side of the carb that works like a fuel pump to pull the gas up in the carb you can try replacing that. If that does not do the trick yes you need to remove and clean the carb. The foam in the tank is not a filter it is just to keep the gas from splashing around to much.
Take the carb and tank off as one unit then drain out of fill cap. A little difficult some hard the get to bolts. As for clean or replace is time or money. New carb you know will be good if you try to clean it and don't get it clean you will have to take it off and try again.