The creature card you return with Cauldron's Gift may be one that you just put into your graveyard with its adamant ability. This is because Cauldron's Gift doesn't target the card that you'll return.

If an instant or sorcery spell has an adamant ability, you perform the spell's instructions in order. You don't perform the adamant instruction before the spell resolves or before any other effects printed above it.

No player may take actions between the time you put cards into your graveyard, the time you choose a creature card to return, and the time you return that card to the battlefield.

If an effect allows you to cast a spell without paying its mana cost, you can't choose to cast it and pay unless another rule or effect allows you to cast that spell for a cost. Similarly, you can't waive a cost reduction unless that effect says you may.

If the creature would enter the battlefield with no +1/+1 counters on it, it enters with one +1/+1 counter on it.

Adamant effects check what mana was actually spent to cast a spell. If an effect allows you to spend mana “as though it were mana” of any color or type, that allows you to spend mana you couldn't otherwise spend, but it doesn't change what mana you spent to cast the spell.

If you copy a spell that has an adamant ability, no mana was spent to cast the copy at all, so that ability won't apply.