On the understanding that something will be done as someone has promised. He agreed to overtake the project on top of his existing work on a promise of a pay raise.We decided to let it go without punishing John on a promise that he will not attend any of these parties ever again.See also: on, promise
on a promise
(of a person) confidently assured of something, especially of having sexual intercourse with someone. informalSee also: on, promise