Most push mower style engines have a molded in fuel filter inside the tank. The surging may not be directly related to the fuel filter in the tank. More likely old fuel residue or other things have made the fuel passages inside the carb partially clogged. Needs a good carb cleaning. Or in the case of most the GCV series just easier to replace the carb. And some of the carbs for the Honda GCV engine are under $20 for the OEM carb so don't even look at aftermarket Amazon or Ebay carbs.
I find it better to clean the carb than take a chance on an aftermarket that may be flawed. They are pretty easy to clean. I have a good success rate. The bowls and gaskets can be purchased separately if they are damaged beyond reasonable repair.