If Harry Potter had a child while he was a Horcrux, would this child have some of Voldemort's powers?
(For instance, would it be a Parselmouth?)
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According to the answers here and here, Harry looses his Voldemort related abilities, including the ability to speak parseltongue, at the time that the part of Voldemort's that lived within Harry was destroyed. This being said, these traits were not part of Harry's gene pool, and therefore would not be passed on to his children.
If Harry had a child while he was a Horcrux, the child wouldn’t then inherit any of the Dark Lord’s powers from him, since Harry himself only had them because a piece of the Dark Lord’s soul was in him. The child wouldn’t have a piece of the Dark Lord’s soul, so wouldn’t have any of his powers.
“Part of Lord Voldemort lives inside Harry, and it is that which gives him the power of speech with snakes, and a connection with Lord Voldemort’s mind that he has never understood.”
- Harry Potter and the Deathly Hallows, Chapter 33 (The Prince’s Tale)
Once the piece of the Dark Lord’s soul is gone, Harry no longer has his powers, further proving that his powers in Harry are directly linked to the soul piece, and won’t exist in Harry without it.
Nigel: Can harry speak parseltongue when he is no longer a horcrux?
J.K. Rowling: No, he loses the ability, and is very glad to do so.
- J.K. Rowling and the Live Chat, Bloomsbury.com (July 30, 2007)
Since the soul piece would still remain in Harry if he had a child, and the child wouldn’t have its own piece of the Dark Lord’s soul, the child also wouldn’t have any of the Dark Lord’s powers.