have a late night
To go to sleep later than one typically does. She must have had a late night last night considering how she yawned all through that meeting.
have an early night
To go to sleep earlier than one typically does. You need to have an early night tonight so that you're rested for your exam tomorrow.
have an early/a late ˈnight
go to bed earlier or later than usual: I’d better have an early night if I want to get up at 6 o’clock.I’ve had a lot of late nights recently.