Background checks can only be deleted while they are in the "Invited" status. To delete, navigate to Contacts > Screenings from the left-hand menu.

Then locate the background check you want to delete and choose "Delete" from the menu at the end of the row.

Once the background check status moves to "Pending" or "Complete," it can no longer be deleted. Instead, you can archive them. That will remove them from the list by default (can be brought back by updating the table filters).

To archive, it's the same steps as for deleting, but you'll see "Archive" in the drop-down menu instead.