Procedure to get rid of fat pockets on sides of mouth? (Photo)

Thanks for including a photo! The aging you are seeing here is a result of the ligament separating your cheek from your chin. Though the cheek falls with time, the chin tends to move less. This creates a prejowl sulcus that can pull down the corner of the mouth. This area can be addressed with facelift surgery. In the interim before surgery, you may consider using some filler to that area to help with the contour. Restylane Refyne is the filler I use most commonly for this issue. Best of luck!

You can try fillers or skin tightening with micro-needling with radiofrequency like Secret to help tone and tighten your face and neck. It works very nicely with little downtime.

Based on your photo, I would consider a conservative liposuction performed through a small stab incision at the jawline or even within the mouth, to reduce the fat deposit. The risk is that the overlying skin could be slightly looser, increasing the depth of the groove in front of the fat bulge. That doesn't always happen, but it is a risk. Of course, with a subsequent surgical lift, the skin would be advanced up and backward, tightening and smoothing the area. An alternate approach is to leave the fat, but inject filler into the linear groove to plump that up which would camouflage the bulge by blending it into the side of your chin.