Here is a link to view some Briggs Pulsa jet carbs and tips for repairing.
https://outdoorpowerinfo.com/repairs/
One of the holes under the pump diaphragm connects to the pickup tube.
If you have air supply you can try blowing air through the pump hole or up through the screen.
The other option is to punch a hole in the screen and run a cleanout wire up the tube, then use a piece of nylon stocking or I've used gas filter socks, type used on in tank fuel pumps. I just used a piece of solid copper wire to tie the filter sock in place.
That carb was probably a 296784 or 393976 if a 2 quart tank or a 296785 if a one quart tank.
Those type were also used on lots of Briggs 3.5 hp vertical push lawn mower engines.
Since it's an edger you could get by with the 1 quart tank and carb.
Those tanks sometimes clean up really good because the were coated well inside.
I've found the pickup tube clogged hard usually within 1-2 inchs of the pickup screen where old gas has evaporated and left crud behind in the tube.
It's best to always install a new pump diaphragm which is readily available and can usually be found in a bulk package of 5 or so dirt cheap.