Some spells, such as the Blizzard (Ice Magic), give you a chance of Weakening a monster.
When a monster is Weakened, it turns white. A snowflake icon and a turn counter appear in front of it. The counter shows how many times you can draw a card before the Weakening effect ends.
To complete a challenge that asks you to “defeat X Weakened monsters", you must kill a total of X monsters WHILE they are still Weakened. 
Note: If the monster is no longer Weakened when killed, then it WON'T count for the challenge.
Learn more about: How do I Weaken a Monster?