where did the phrase A lick and a promise come from?

Answer 1

A Lick and a Promise:

Meaning

A cursory effort, for instance at painting or tidying up. It alludes to the perfunctory washing performed by children.

Origin
This is colloquial English and is first recorded in print in Walter White's All round the Wrekin, 1860:

"We only gives the cheap ones a lick and a promise."

Answer 2

I think my Dad started that when he used to say, "Give me a lick of your ice cream cone, I promise I'll give it right back" but he never did.