I have these "fat pockets" on the sides of my mouth that cause my nasolabial folds to appear more pronounced. What is this?

Treatment of ‘chubby cheeks’ cannot be achieved alone with buccal fat pad removal; a fact that I encounter time and time again when seeing patients in consultation for lower facial sculpting. My approach is to use a combined protocol that redefines aesthetic contours of the face. This includes the following: 
  
Buccal Fat Pad Removal Non-surgical Muscle-Fascia Tightening Venus Legacy or Venus Viva Ultherapy Submental Liposuction or Coolsculpting   
The combined approach is ideal. Buccal fat pad removal is a exceptional procedure that can remove excess fat from the cheek pads. Using an intraoral approach, a small incision is carried through the mucosa to remove a small fat pad within the cheek. Closure of the incisions is minimal with an absorbable suture and patients will experience some swelling post-operatively. We use SwellX tablets post treatment. Once healed, the contour of the cheek is improved and we begin Non-invasive tissue tightening to further enhance results. 
  
Best, 
Dr. Karamanoukian 
Realself100 Surgeon

This is very attractive feature of young perky lips, in my opinion. This may be treated with very superficial, fine cannula liposuction but I would not recommend it in your case. In person consultation with an experienced board certified plastic surgeon is recommended. Good luck.

Thanks for your question and Pictures
The buccal fat pad is a rounded mass of fat in the middle of your cheek. It’s located between facial muscles, in the hollow area beneath your cheekbone. The size of your buccal fat pads affects your face shape.
Everyone has buccal fat pads. However, the size of buccal fat pads can vary greatly.
If you have larger buccal fat pads, you might feel like your face is too round or full.
You can be a candidate for a Buccal Pad removal surgery.
Best of Luck