Can you cook with a partially healed open wound?
According to the World Health Organization, it is technically possible to catch certain diseases by handling raw meat with cut or abraded skin. However, that example was a pig disease only occurring in a specific region.

I could find no references to similar disease transmission from chicken, and you definitely won't get food poisoning, which relies on long population multiplication times. As long as you practice basic handwashing, care for your cuts normally, and don't live in a region with notable chicken diseases, you'll be fine even in the case of an open cut.

Just try not to bleed in the food, please.