I recommend speaking with your doctor
about this concern. A bladder infection and a cut on the labia should be unrelated. A bladder infection is due to bacteria in your bladder or urinary tract causing infection and irritation. This can manifest with needing to pee frequently, pain or burning with peeing, pain in your lower abdomen, foul smelling urine, and in serious cases, fevers, chills, pain in your back or sides, and blood. This can be diagnosed by a urine analysis and culture of the urine. The cut should cause more localized symptoms - pain at the site and if it's deep maybe some swelling. You could have some pain and burning with urination if your urine is irritating the cut, and it's possible that these are your symptoms rather than a true recurrence of your urinary tract infection
. On the other hand, if you do have a true urinary tract infection it would be possible though unlikely to get a bacterial infection of the cut as well. I would make sure to keep the area clean (no need to use soap which can be irritating) by wearing clean underwear and making sure it's not too moist. If you are having urinary symptoms I would go to the doctor to have them check your urine for an infection and possibly prescribe antibiotics if you need them. The cut should heal fairly quickly, but if it's not improving within a day or two I would also have the doctor take a look and make sure that the area isn't infection.