Generally speaking, any kind of stuffing is dangerous.
1) for health reasons (respiratory complications from fumes, dust and/or bacteria) and the possibility of intestinal blockages if ingested)
2.) It can encourage hormonal behavior (aka, nesting, screaming, feather plucking, aggression and egg-binding in females which can result in prolapse or death).
The bird shouldn't be walking around with its face in the same bedding it poops in.
This is probably nesting behavior and should be discouraged. On a similar note, if you have any huts, boxes, tents or any dark/shadowy places in the cage, remove them as well...That ledge/plastic wedge shaped perch so close to the cage-bottom is very bad because it creates a cave/dark nesting place (again, you absolutely do not want to give any dark spaces to your bird)---notice how your bird chose to nest under it. Take it out if you can and stick to stick-type perches that do not cast such massive shadows.
Does he have any toys/perches higher up in the cage? Does he sleep on the ground even though he has perches (that can signal a serious health issue).
Back to the bedding issue-if you have a cage grate, plain old paper towels can be used beneath it, as can black and white newspaper or butcher paper. If you don't have access to any of these, do not use a liner and just make sure you clean it often.