to use a piece of cloth or a mop to remove liquid from the surface of something

There's milk on the floor over there - could you get a cloth and mop it up? 那边地板上有牛奶——你能拿块布把它擦干净吗？

to finish dealing with something

It took a week to mop up the last of the enemy soldiers (= defeat them). 彻底消灭敌军残部花了一个星期。