Can you put a card with cycling into your graveyard and then exile it for delve in the same turn?

Cycling is an activated ability you can activate any time you could normally play an instant (CR 702.29a). When you cycle a card, you discard it — which puts it into your graveyard — and then draw a card. The card is now simply a card in your graveyard; it has no memory of how it got there.

Delve (CR 702.65a) lets you exile cards from your graveyard as you cast a spell, paying 1 generic mana less for each card exiled this way. There is no restriction on how those cards entered the graveyard, so a freshly cycled card is a perfectly legal target for delve.

The only ordering requirement is that the card must already be in your graveyard at the time you cast the delve spell and choose how many cards to exile (CR 702.65b, 601.2). So you cycle first, and then later in the same turn you cast a spell with delve, exiling the cycled card to help pay for it.

Concrete example: You cycle Lay Claim during your main phase, discarding it and drawing a card. It is now in your graveyard. You then cast Treasure Cruise (delve). You exile the cycled Lay Claim from your graveyard, reducing Treasure Cruise's cost by 1. This is completely legal.

One caveat: if something like a replacement effect (e.g., Rest in Peace) exiles cards as they enter the graveyard instead, the cycled card would never reach the graveyard and could not be used for delve.