The card says When Played: Spend 1 influence to destroy 1 opposing unit.  OK, I am attacking (or being attacked) by an Enemy event card.  If I play this card, and spend my influence, can I immediately destroy it?  This is how I was playing it in the solo game, but this seems extremely over-powered. Am I interpreting this correctly? 

Actually I was wondering the same thing.  This also pertains to the Elves influence and the archer "untap" strongholds/ cities for influence.  When can the elves spend influence and tap more strongholds to trigger abilities during combat?  Is it outside the spend influence step?  Can you "hold" influence for later in the combat step after you exhaust for influence earlier in the turn.  When we were playing the game it seems the elves can snag up all the high cost neutral cards real quick with archers and a few strongholds.

I think you will find that 'Enemy' event cards are not counted as units and so the effect wont work.

As soon as the Elf Archers are played in combat, the owner can refresh one of its exhausted strongholds or cities.  This, and other influence cards, can be used during that combat to trigger any effects that require influence.  They cannot be used to gain cards during the combat, but must wait until the combat is over and you can make other actions.  Cheers!

 Yep - the rules clearly state that an "Enemy" is not a "Unit" and vice versa.

Think of Enemy cards as another player.  You can't use the destroy effects on them.