To provide a person with another opportunity or chance; not to expect too much work from; not to expect someone to believe.
Command forms are most common with this idiom.
Oh, Jim, give me a break! That’s a terrible excuse for being late.
The driver pleaded with the police officer to give him a break and not issue him a ticket for speeding.
When the students heard how much homework the teacher wanted them to do over the holiday, they begged, «Give us a break, Professor Doyle!»