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If acid spit is activated with 1 MP one round, then the next round it is used again on the same opponent who has not successfully done a Move to stop the acid damage, does the per-round acid damage go up? In other words:
Round 1: Acid Spit on Foe, they take 1 point damage. Foe fails Move on their turn or chooses to keep attacking mindless of the damage.
Round 2: Acid Spit on Foe again, they take 1 point damage. Foe take one point of acid damage from round 1 acid. Foe still doesn't succeed on Move or doesn't try Move.
Round 3: Swing bat at Foe and miss. Does Foe now take 2 points of damage on their turn from recurring acid damage, or just 1?

We played where the foe did take two on round 3 because the second acid spit drenched them more and so they took more recurring damage, but not sure if that is the right call.

You're right, your opponent has more acid spit on them and thus will take more damage each round. It will only require one Move roll to remove all acid.