Would a powered USB extension cable cause issues or improve a USB microphone when trying to record high quality audio?

As in any live recording applications, the more length you add to cables, the more noise that cable tends to pick up. Try to keep cable lengths to a minimum. If that is not possible, give it a shot and see how the recording sounds. You may want to try an XLR microphone application as XLR cable are shielded to reject unwanted noise*.

*XLR cables may still pick up unwanted noise when running them at long lengths.