Will Ezreal Prodigy make the additional cost of Pyke Dockside Butcher free?

Yes, Ezreal, Prodigy will reduce the additional cost of playing Pyke, Dockside Butcher to zero.

According to the text of Ezreal, Prodigy, "Optional additional costs you pay cost [1] or [A] less." Pyke, Dockside Butcher features an optional additional cost to play him: "You may pay [C] as an additional cost to play me."

Because [C] (Chaos/Purple rune) is a type of rune/power cost that can be reduced by the [A] ("any domain") discount provided by Ezreal, and because the discount applies to the optional additional cost component, the cost is reduced. Per Rule 356.4.f, discounts can reduce additional costs to 0.

Therefore, if you control Ezreal, Prodigy, you may choose to pay the additional cost for Pyke, and the [C] cost will be reduced to 0 by Ezreal's ability.