When an effect lets you play a card, does that count against your allowed number of card plays?

When an effect lets you play a card, does that count against your allowed number of card plays?

No. The number of card plays on your Presence track only affects how many cards you can play as part of the "Play and Pay For Power Cards" action in the Spirit Phase. Cards played any other way (usually in the Fast Phase when instructed to do so by a Power) are in addition to those limits.