Yawgmoth's Will

If you cast a Buyback spell, then there will be two effects trying to replace where the card goes. You get to choose if the Buyback returns the card to your hand or the card gets exiled.

It will exile itself since it goes to the graveyard after its effect starts.

If an effect asks you to discard a card, you can’t “discard” something that is in your graveyard. Those cards are not in your hand. Thus, Cycling abilities of cards in the graveyard can’t be activated.