It's probably an overly lean fuel mix. You can adjust ignition timing somewhat by moving the coil and changing flywheel keys. Cam timing is pretty much non-adjustable except for performance applications and then you can index the cam, but this requires some modifications.
Backfiring through the carb is indicative of a lean mix probably 90% of the time with a valve sticking or out of adjustment 10%. If yours has an adjustable carburetor, I'd try resetting the jets first followed by rebuilding the carburetor and adjust the valves last. I think the lash is .006" for intake and .008" for exhaust although the guys who race the Hondas will tighten that up a good bit to get a faster opening valve.
I'm sorry, I thought you meant it was backfiring while running just after started.
I would check my flywheel key first. If the key is bad and the ignition too advanced, it can do this.
You'll need to check the valve lash with a feeler gauge. With the piston at Top Dead Center the lash should be about .006" for intake and .008" for exhaust. You may very well have an intake valve that has a seating problem. If the valve lash is correct and the key in place, then I'd probably go ahead and pull the head and inspect the valve faces and seats.
Honda lists something like 110-180lbs with the compression release not functioning. I would want to see probably 75-85 minimum but even that may be high depending on how fast the engine is spinning. I've had engines with 50psi on the release that ran fine. A compression test is more useful on a twin, or larger, because then you have something to compare it to. A more useful test is a leakdown test, but in your case I don't think I'd do it, I think I'd just move on to pulling the head.
So a compression test alone might not be able to show you a valve that's not seating. I had a Kohler or a Briggs, I can't remember which, last summer that had a valve seat that had come loose and didn't show up on a compression test or a leakdown test at TDC. I was so sure that the valve was leaking that I pulled the head and there was a valve seat, on the intake side I think, that would just barely slip down when I compressed the spring. This engine would start and pop and buck, but the basic tests just wouldn't reveal it. Sometimes you just have to start removing parts, although that is my last course of action.
You're on the right course. I think if you inspect the valves and seats, head, head gasket and engine deck you'll find your problem. It could be nothing more than carbon built up or it could be a seat loose. I think close inspection would be the diagnostic tool to turn to with your symptoms. Thanks, PK.