" Could this be a fuel pump issue?"

Could be - do not continually spray Starter Fluid in carb.

You can check this by rigging a temporary tank so that it will gravity feed into engine.

Check for cracks in the vacuum line running from the bottom of the fuel pump to the vacuum take off on the engine block.

Walt Conner

the information from Walt applies.
Here are a couple of additional ideas.

If the fuel line from a rear fuel tank has cracks, the engine fuel pump may be drawing in air. The areas with clamps should be examined.

If the mower has experienced outside storage, the fuel tank may have some water. Water may also be in the spare fuel can.

The 10% ethanol fuel runs ok when it is fresh, however, if it is stored a long time, the alcohol may separate from the gasoline and absorb water.

I find that often the main jet gets clogged on these carbs because the main jet feeds from the bottom of a well in the carb bowl.