Possible Cain(e) connection.

If Edward's mother was Elizabeth Callow 1797 Malew she is likely the d/o John Callow & Catherine Caine who md 1789 Malew.

HOWEVER, I'm having trouble accounting for Ann Callow Gawne b 1791 wife of Thomas Gawne in the 1851 Liverpool census

I don't see a baptism for an Ann Callow around 1791 in the probably appropriate parishes.
So the above is the wrong Eliz Callow OR
Ann is her sister b 1791 but not christened or record lost
Ann was Ann ? md to a Callow, perhaps sib of Eliz and was her sister-in-law as md to a Callow not a birth Callow
Or many other possibilities

DJD