Peter Kennedy includes the song, under the title "The Foggy Foggy Dew", in his _Folksongs of Britain and Ireland_ (Oak, 1975). The version he gives (he lists a lot of others) resembles the one quoted above from Martin Carthy, Cyprus cat & all. (I presume, however, that the "cod" offered was actually a cot; Kennedy has "crib".)

In the introduction to the chapter containing the song, Kennedy says:
"James Reeves, in trying to discover the significance of the title, suggests 'fogge', the 'Middle English for coarse, rank grass of the kind which grows in marshes and bogs where the atmosphere would be damp and misty', and this, as in _Rolling in the Dws_..., would represent maidenhead, and the dew would imply virginity or chastity. 'Foggy Dew' may be an English tongue's best attempt at the sound of the Gaelic, and derive from 'Oroce dhu' meaning a black or dark night. Robert Graves proposed a theory that it stood for the black pestilence of the church and that the girl was really being protected from entering a nunnery. There seems to be no end to what can be interpreted from the lines of folksongs."
For this last statement there seems to be plenty of evidence.