1. with a vengeance : 猛烈地；
名词
1. 复仇;报复;报仇
Vengeance is the act of killing, injuring, or harming someone because they have harmed you.
2. 过度地;猛烈地;激烈地
If you say that something happens with a vengeance, you are emphasizing that it happens to a much greater extent than was expected.