Tuned exhaust? No, these machines are just very basic in that they do have some kind of muffler on them, not much else can be said for the exhaust.
A crack in the exhaust system is not going to have any appreciable effect on the performance of the engine.
Typically, an exhaust system on any outdoor power equipment will not affect the performance of the engine unless it sheds some internal baffling partitions which then obstruct the muffler outlet port, causing a lot of back pressure, which in turn prevents the engine combustion chamber from purging efficiently during the exhaust stroke.
Now, it is very possible that the popping sound is caused by the engine exhaust being fairly high in unburned fuel.
The split or crack might be allowing just enough atmospheric air to be drawn into the hot exhaust to support a flash burn inside the muffler.
Normally, the unburned fuel in the exhaust stream is unable to burn inside the muffler because there is no air in the mixture, and because the gases are continually blown right on through the muffler and into the open atmosphere where they instantaneously cool BELOW ignition temperature.
But a breach in the exhaust system can act as an entry point for air to be drawn into the exhaust by the inertia of the exhaust gases traveling through the exhaust. The gases are in motion and have inertia. When the exhaust valve closes, there is no longer any positive pressure pushing the exhaust gases through the exhaust system, but the gases inside the pipe and muffler want to keep moving because the have momentum.
It is the forward momentum of the gases that tends to create a negative pressure (vacuum) in the exhaust system. Because of the negative pressure in the exhaust system at this time, a crack or other breach will allow a small amount air to be drawn into the exhaust system. Because the exhaust system surfaces are still hot enough to permit ignition, and because there is now enough air to also support combustion, the hot fuel component in the exhaust will light off and cause a small pop.
That may be what you are hearing.
Yes, it is necessary to run some kind of exhaust system on an engine, at least it needs to have a short length of pipe extending beyond the exhaust port of the head.
Reason for that is because without SOME bit of back pressure to slow down the exhaust gases leaving the combustion chamber, uneven heating of the exhaust valve and seat will occur. The long term effect of operating in that condition will lead to uneven wear of the valve and seat due to a slight tilt or warp that the valve will take on because of extra heating on one area of the valve.
I doubt anybody could put up with the racket a one lunger engine would make without any exhaust system, at least not for very long.

Thanks mownie for that. Very interesting.

How do you think JB weld would hold up on a muffler. They say it is good to 550F? I also have some high temperature gasket maker they say is good to 650F that might seal up this crack. The crack is about an inch long and kind of hair like in its width.

Be best to remove the pipe and have it welded. That's if the muffler is still in good condition assuming it's welded to the pipe.
Mike

No kind of RTV is going to resist the conditions it will be exposed to in that venue.
RTV used as a sealant (as in "caulk") has very little structural strength, and would succumb to the high temps plus pressure.
The exhaust pipe can easily reach temperatures of more than 900 degrees F at times. I have seen exhaust pipes on perfectly good engines glowing a dull red in dark of night conditions, which by my estimation (comparing to a color vs. temp chart) was somewhere in the neighborhood of 1,100 F.
As mla2ofus suggested, welding is the route to take in lieu of buying new parts.

Depending exactly where the crack is on the muffler. (Read what is screewed into the block) you may be able to pick up a replacement at your local big box store. I have not priced them recently, but I believe the cost of the item is less than the cost of messing around trying to fix it and the cost of the materials you will need to try to fix it.