Fat around corners of mouth (perioral mounds or buccal fat)? (Photo)

It's a pity that you didn't show photos of the entire lower face - it's hard to judge based on these photos, but it's possible that the solution for you would be to reduce the bichat pads.

Those are subcutaneous collections of fat known as perioral mounds. They can not be successfully treated by a buccal lipectomy which may make them look more pronounced as the cheek above it gies more inward. They require small cannula liposuction from a small incision inside the mouth to be removed. Weight loss will not reduce their size.