Cooking with a cut or burn on my hands
If you're really worried about it, put some superglue over it. It actually works really well...Just don't stick your fingers together. Regular superglue can irritate your skin, so if you have time to plan ahead, buy some Dermabond, which is a cyanoacrylate surgical glue...They use it in place of stitches sometimes. Works great. Your wounds will NOT seep. Period.

Honestly though, people get cut all the time in professional kitchens, and I've never seen someone do more than slap some glue or a piece of tape over it, and keep going. You can't just quit work for a couple of days without throwing the whole business into chaos. It's a rough trade.