The phrase suggests a task is done incompletely, with an intention to finish it later. I only had time to give my essay a lick and a promise before the submission.
Pro Tip 2/3
Not for Formal Use
It's casual and may be inappropriate in formal or professional communication. You wouldn't 'give a report a lick and a promise' in a business setting.
Pro Tip 3/3
Suggests Rush
Implies something was done hastily, often due to a shortage of time. Sorry for the sketch, I could only do a lick and a promise during the meeting.