Being "On a promise" means that (usually) a husband is expecting to get home and have and "early night" (Nudge, nudge, wink wink...). In other words he is on a promise that his wife will let him have sex (and presumably this is a rare ocurrence), so he is wanting to get home early from the pub, instead of staying until throwing out time with his friends as would normally be his habit.
Quack!
GtD.