A cleave cost is an alternative cost that's paid instead of the spell's mana cost. Casting a spell for its cleave cost doesn't change the spell's mana value.

If you cast a spell for its cleave cost, that spell doesn't have any of the text in square brackets while it's on the stack.

You can't cast a spell for both its cleave cost and another alternative cost. For example, if an effect gives an Alchemist's Retrieval in your graveyard a flashback cost of {U}, you can't cast it from your graveyard for its cleave cost.

If an effect allows you to “cast a spell without paying its mana cost,” you can't cast that spell for its cleave cost.