Richard Runciman Terry traces this capstan shanty to folksongs of the Northumbrian coast (Northeast England and Southeast Scotland). He explains:
kittled = tickled
gairdle cake = griddle cake
Singin' Hinnies - a species of teacake, usually plentifully besprinkled with currants
Stan Hugill adds that there were variants from other parts of Britain. The timber-droghers had a related version known as "My Bonnie Hieland Lassie", and the Appalachian-American variant is called "Willie Can She Cook?".
This is one of the very few shanties which can be lead by two shantymen -- one asking the questions and the other responding.
The song is part of the general folk song tradition; see, for example, Ed McCurdy's recording on The Folk Song Tradition.