1.
If you ~ a liquid, you clean it with a cloth so that the liquid is absorbed.
A waiter mopped up the mess as best he could...
When the washing machine spurts out water at least we can mop it up...
Michael mopped up quickly with his napkin.
PHRASAL VERB: V P n (not pron), V n P, V P
2.
If you ~ something that you think is undesirable or dangerous, you remove it or deal with it so that it is no longer a problem.
The infantry divisions mopped up remaining centres of resistance...
PHRASAL VERB: V P n (not pron)