Can a clogged screen in the carb cause backfire issues or is my time better spent looking elsewhere?

What kind of engine? Backfiring in our engines is usally a timing issue or low voltage. One loud backfire could be blown reeds.

As the british would call it "banging in the silencer" can indeed be caused by an over rich mixture, especially with sudden reduction of throttle. Ignition timing can cause it also.

Bill

Not usually a fuel problem, more probably carbon build up on the plug internally or to much advance on the timing.