One source says "...the hasty performance of a task, or something not done properly, also (originally) a hasty wash - the expression is probably from the first half of the 20th century, possibly around 1910-20's, and originally meant a quick or superficial wash (usually of a child's face by the child). It was certainly well in use by the 1930's for the original literal meaning..." (Source: The Free Dictionary)
Thanks, Clanad. You have retrieved for me a very early childhood memory of my mother wetting the corner of a handkerchief with saliva and rubbing a dirty mark off my face!!!! Completely (and, probably) best forgotten!
We used to have 'a lick and a promise'. I assume (now) that it means a quick wash today and the promise of a bath/shower tomorrow. It was always the night The Goodies were on - that shows how old I am!!!