Is it permissible for Zakat and orphan sponsorship to be given to the child of adultery? Thank you.

The child of adultery is innocent and not held accountable for the sins of his parents, as Allah Almighty says: "And no bearer of burdens will bear the burden of another." It is permissible to give him zakat if he is a poor, free Muslim. Giving to him may even be better than giving to others if he is in dire need and lacks care.