To obtain a Damaged Mask, you need to find and kill Hilichurls at any level. Upon eliminating these standard enemies, they have a chance to drop Damaged Masks.
You can get these from any of the Churl Family, such as Samachurls, Mitachurls, and Lawachurls, but not as often. This means killing bog-standard Hilichurls is your best bet, as they are the fastest and easiest to kill.
The fastest way to get around is to use the Waypoints, kill everything in sight, and then move on. The two images below are where you can find Hilichurls in both Mondstadt and Liyue.
Just click on them to enlarge them.
Thanks to Genshin Impact Interactive Map!
- Type: Common Ascension Material
Damaged Mask is used for the following Character Ascensions.
Each Common Ascension Material has three tiers; these are the next tiers after Damaged Mask.