Slitherine

If I occupy the enemy's supply hex (with the circle split in two ) what effect does that have on the enemy? Is there only one per map?

If you occupy a hes it is your hex not the enemies.
The effect depends the map.
Map edges provide supply, so often it will have no effect.

If you encircle the enemy and there is a supply hex inside the encirclement the enemy is not affected by the encirclement.
Once you take that hex, the encirclement effects start.
This is a big deal on the W****w Map.

So it is dependent on the map and your disposition of units.