Not 100% on the resulting color but I have heard that barring genes are usually dominant (there are different types of genetic combinations that result in barring). However, the naked neck genes are dominate so if you mix your game hen with the naked neck, the result will likely be naked necked.
Sorry that I can not provide more info on the subject: I am not a professional breeder.
From the Houdan rooster and that hen, expect black mottled chicks (it's a recessive gene, so both parents are pure for that) with white skin and partial crests.
The Naked Neck rooster only has one copy of the barring gene. He's got a bit of gold leak indicating both silver and red/gold base color genes. The gene for the naked neck is dominant, but he might not be pure for it. Expect about half to inherit barring, half the chicks to be silver base color, and half to be red/gold base color.