For Ezreal - Prodigy, what is considered an extra cost that his effect can reduce?

Ruling: Optional additional costs are ones that use the words "may" and "as an additional cost" (or "may" and "as a cost to play"). Repeat and accelerate are the two most common examples.

Nuances:

• Equip is not an optional additional cost; it's the cost of an activated ability
• Hidden is not an optional additional cost; hiding a card is a game action with a cost, not an additional cost to play a card or effect
• Hard Bargain's "unless its controller pays 2" is not an optional additional cost; it's paying to get a separate effect (not countering your spell)
• Deflect is not reduced by Ezreal because it's not optional
• To be an optional additional cost, it must be both optional AND additional (not just the regular cost of playing the card)