Sorta like a very mild backfire, originating from the carburetor area?.... NOT a loud shotgun like blast like one would hear if blasting out the exhaust?

That would indicate (usually) a lean running carburetor.

Even though the engine is warm after a 10 minute run and this mild backfiring exists... If you shut down the engine, pump up the fuel primer bulb, then restart the engine... does the engine run okay again for that 10 minute period OR does the backfiring start immediately?

Does the problem cease to exist if you constantly pump the fuel primer bulb, acting as a manual fuel pump?...... If so, check for a slight fuel leak at some point between the pump and the fuel supply. A fuel leak would by the same token result in having the pump draw air... however to run normally for 10 minutes before acting up, that would be a very small leakage area.
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Also, a extremely small fuel restriction such as a Anti Siphon valve just starting to act up (sticking) might do that... worth looking at (as follows).

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(Fuel Anti Siphon Valve)
(J. Reeves)

Many of the later OMC V/6 engines incorporate a fuel restriction warning via a vacuum device attached to the powerhead. If the engine overheats, or if you have a fuel restriction, the warning is the same.... a steady constant beep.

NOTE... Only the V/6 & V/8 engines have the above "Fuel Restriction Warning". The warning horn will not sound on the other models.

The fact that a engine is not overheating, but the warning horn sounds off with a constant steady beep, and that the rpms drop drastically would indicate that the engine is starving for fuel due to a fuel restriction. Check the built in fuel tank where the rubber fuel line attaches to the tank fitting. That fitting is in all probability a "Anti Siphon" valve which is notorious for sticking in a semi closed position. It will be aluminum, about 2" long, and the insides of it will consist of a spring, a ball, and a ball seat. If this valve exists, remove it, knock out those inner components which will convert it to a straight through fitting, then re-install it. Hopefully that cures the problem.

The above procedure will cure a restriction problem with the anti siphon valve as stated. BUT, it may also allow fuel to drain backwards to the fuel tank when the engine is not running (siphoning backwards) due to the fact that the carburetors/fuel pump etc are higher than the fuel tank. This condition is not an absolute as the valves in the fuel primer bulb usually prevent this backwards siphoning problem. However.... if this does take place, the cure would be to install a new anti siphon valve.

NOTE: There has been cases when the output valve in the fuel primer bulb would come apart, and the inner portion of the valve would actually reverse itself and be drawn back into the primer bulb's output valve body. This in effect would create a shut off valve and result in a fuel restriction. If this is the case, you should be able to feel something laying in the bottom of the primer bulb when held horizontally.

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