In FCP you can try one of the lowpass filters, which allows the low frequencies to pass through while sloping off some of the highs, or you could even try the vocal de-esser, which also gets rid of some of those high frequencies that come with static and clothing sounds. However, since I don't know exactly what the unwanted sounds are like, you'll have to play around. My best suggestion would be (failing re-recording, which it sounds like you can't do) using a 3-band EQ and trying to isolate the bad frequencies as much as possible and notch them out without affecting the voice too much.
Also, if you can get your audio into Soundtrack Pro or Pro Tools, you'll have much more control over it than you would within FCP, and then you can just import it back in.
You won't be able to remove the extraneous BG sounds completely, but you can certainly dampen them with enough tweaking of filters. As far as the music track muffling them, that depends on what kind of track you're using. Certain types of music might make them sound worse, depending on conflicting frequencies, but if you choose the right one it could help out a little bit as a bandaid.
If you can record this statement again if at all possible, that would be the absolute best way to go. Of course I understand that might not be an option.
Hope this helps,
Joni