See my answers above in red.1. "Under no circumstance, he would break the promise to pay back the money."
Does this sentence mean he will return the money or not?
I think maybe I change it like this would be easier to understand.
"Under no circumstance, he would break the promise of paying back the money."
The sentence is incorrectly written. It should be "Under no circumstances would he break his promise to pay back the money." The correct word is "to".
2. "----When did you find your bag?----It might have been at Christmas, when John gave me the book."
This is correct, but if you omitted "at", it would still be understood. Note that I have added a comma after "Christmas" so that it means "I might have found my bag at Christmas, at the same time that John gave me the book". The suggestion is that I can't exactly remember when I found my bag.
"----When did John give Mary the book?----It might have been Christmas when John gave me the book."
This is correct without "at".
In order to understand the difference between these two answers, I created two questions to match the two different answers , am I right?