Can a small cut on my foot (from a tree branch) hurt my baby if it gets infected?

Possible. Another words you are pregnant. If you get an infection and the infection spreads you can get an infection in your blood etc etc...Best thing to do is make sure you dont get an infection and dont worry. Apply topical antibiotic and keep an eye on the wound.
Cut. Absolutely . See your podiatrist or pediatrician for assistance - little one's need it faster as their immune systems are immature.