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Does Harmful Aura do its damage as soon as a PC comes into make range?

And on following turns, does it do its damage at the start of the creatures turn or the start of each players turn?

We've always read it as you taking damage when you spend at least a part of your turn exposed to the aura:

If you are standing inside the aura at the start of your turn, you are affected by the aura.
If you move into the aura on your turn, you are affected by the aura.
If the creature moves into melee range with you on its turn, you don't take the damage at that point.

Symbaroum is a pretty simple system, and tends to be player-centric, so the simplest solutions usually work best with it.