The Roud index lists 2 titles with Southampton in the title.

1. A fragment collected by Baring Gould (3v) - In Southampton City, scan of words and tune is in the entry.

2. A Most Strange and Trew Ballad of a Monstrous Child Borne in Southampton...1602... in Clark: The Shirburn Ballads, which you can find at archive.org. The child was born with extra mouths and ears, a sign of God's anger!

There are ca225 entries for songs actually collected in Southampton

Mick