Is there a way to recover cards from the mana section?
Certain cards will let you move mana cards from the mana zone to other zones including the play zone. As an example, the Greed God Moloch allows you to play minion cards from your mana zone as if you were playing them from your hand (except you can't use their summon effects).
Can you exile your own minion/fate/relic cards in the mana area into your hand to then be played on the "play zone"?
Any cards in the mana zone are mana cards; that is, while in the mana zone, they lose their minion/fate/relic status and are simply mana cards. Cards with the exile ability will specify valid targets. It will say something like, "Exile target relic," "Exile target abomination," or "Exile target minion." So mana cards would not be valid targets for those particular abilities.
Even so, exile abilities can target cards in the Mana Zone. The Screamroller, a Deceit Minion, has a trigger ability that will allow you to exile one of your own mana cards to your hand.