The wording on this card almost makes me angry.
I don't know if I'd go that far (angry?). The wording seems clear enough to me, though I'm not sure why it goes with the convoluted "this mage...they..." rather than the more straightforward "you" (equipment is always controlled by the mage that has it equipped, so "you" means the same thing in this case).
The ability of this equipment triggers at the end of the mage's action phase. It only triggers a single time at the end of each action phase.
Well, that is a bit less clear from the wording of the card. Still, all in all I don't foresee too many other clarifications that need to be made.
I suppose a simpler wording might be "At the end of your action phase, you may target one creature in your LOS and pay 2 mana to place a Burn condition on that creature if it has no Burn conditions and at least one revealed curse you control attached to it."