To clean or dry a surface by wiping it with a mop or cloth. After the spill, she had to mop up the water from the kitchen floor.
To finish or complete something, especially by dealing with remaining details or tasks. The team stayed late to mop up the final details of the project.
To eliminate or deal with the remaining parts of a problem or situation. The police were called in to mop up the last of the resistance in the area.