Carb needs cleaning. There is no fuel filter fitted as standard on the Classic thus anything going in the tank will go to the carb. The injector hole is tiny within them. Depending on your model you may have a Delorto or a Tecumseh carb. The bowl will just need a rinse but you should use a small splinter of wood to clean out the fuel jet. Don't use anything else as this will alter the jet, it's delicate.
You may already know this so I apologise if so.
I have taken apart both types of carbs to clean on several models. It's amazing what a difference it makes.
MowerMan
August 2011