I got a small cardboard cut on the top of my thumb 3 days ago. It's not bleeding or anything but the cut hasn't fully healed. Technically this question is after the fact since I just cooked some chicken so I guess what I'm asking is is what I did safe or will I get sick from this?
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.
Handling raw meat with a cut stings (not as badly as lemon juice on a cut). You might want to cover it for that reason alone, as well as to avoid the (tiny) risk of infection entering the cut. In some places on your hands waterproof plasters either don't stick well, or make handling things awkward, so there's a tread-off. In any case you should wash your hands well with soap before and after handling raw meat.