Is this pattern normal for this rifle? If it is, then the problem is most likely a mechanical one. Two distinct POI means that either the barrel or the scope internals are moving between two distinct locations.
This is often a scope issue. I don't know the brand of scope, but Handi owners often buy inexpensive scopes to compliment their inexpensive rifles. "Cheap" scopes can suffer from poor internals with a reticle shifting between two points randomly depending on recoil forces. Actually this can and does happen with top-of-the-line scopes too, but much less often. Install a different scope and see how the barrel performs. I assume that the scope mounts are tight and are not shifting.....
The simplicity of the Handi makes barrel shifting less likely but possible. I would not expect minor shifting of the forend itself to have this much of an impact on POI with a mild cartridge like the .357 Max, but it is possible. Is the barrel tight in the frame when the action is closed?
The rifling type would not be the cause of this issue. The rifle shoots nice, tight groups - it just puts them into two different places. :cry: