My passport is damaged (broken, scribbled, etc.). Can I check in with this damaged passport?

Depending on the status of your passport, you check-in might be denied.
In the event of passport damage, it may be a reason for the destination country's Justice Department to refuse entry. Please get a new passport issued. If you cannot get a new passport due to an imminent departure date, Incheon International Airport will help issue an emergency passport (available on Saturday and Sunday work, but Closed on holidays).