A fire can't have too much air, but certainly too strong of a draft.
In this case though, it sounds more like your logs haven't been fully
seasoned. I used to buy mine at the supermarket. It was supposedly
kiln dried, but half the time it would behave just like you're
describing. Just ruined what could have been a nice evening by the
fire. If you've bought a cord from a local dealer, ask for an exchange
or your money back.
When you cover the opening with plywood you create a stronger draft as
you would get with a woodstove. But if this is what it takes to keep a
flame going then I still say the wood is slightly damp.