What are these fat pockets around my mouth and can they be removed? (photos)

Thank you for submitting your question and photos. I
would not recommend liposuction to the face. The risks outweigh the
benefit. A consultation with a Board certified Plastic Surgeon will give you the best plan. Best wishes

A modest amount of fat grafting to the submalar area just lateral to the nasolabial line will improve this.  It is relatively straightforward procedure that can be performed under local in the office. 

The dark ring is created by a depression at the junction of the mouth and the cheek. A small amount of filler should be able to solve this problem.