Does using neosporin on a cut reduce my chances of a bacterial infection?

Yes. Neosporin has neomycin (kills fungus) and Bacitracin (kills bacteria) in it. Both, however can cause rashes, and you may need to take oral antibiotics if you are already infected. See a doctor if you have questions.
Neopsorin wound. Neosporin is an antibiotic ointment that fights against bacteria so it can be an effective product to put onto a wound that has been cleaned after an injury.