In this song, the band is addressing a groupie, who is the "fox on the run." Apparently she's had more than one go with the band, as Brian Connolly sings that she doesn't look the same - probably a bit more worn from her lifestyle. He also makes it clear that he has no interest in hearing her talk and certainly doesn't want to know her name.
Sweet was produced by the glam rock architects Mike Chapman and Nicky Chinn, who also wrote most of their songs, including all of their hits to this point. "Fox On The Run" was written by the band - Brian Connolly, Stephen Priest, Andrew Scott, Michael Tucker - and included on their 1974 album Desolation Boulevard.
After the album was released, the band parted ways with Chapman and Chinn and produced their own material. Their first effort was a reworking of "Fox On The Run," which was originally helmed by Chapman/Chinn. Defying predictions of disaster, Sweet proved quite capable on their own, and the new version of "Fox," with a bright chorus and bold echo, became a global hit, reaching #2 UK and charting across Europe (it was #1 in Germany, where Sweet was wildly popular). It also became one of the few American hits for the band when it was released in the US as the follow-up to their single "Ballroom Blitz."
You can't copyright a title, but Tony Hazzard, who wrote the Manfred Mann song, thought it was bad form. "'Fox On The Run' is such a distinctive title that any writer worth his or her salt would leave well alone," he said in his Songfacts interview. "They even copied the rising harmonies! As you might gather, I didn't think much of it."