What can I do about the perioral mounds or fat pads by the corners of my mouth?

Fillers to correct volume loss and provide a lift as well as a skin tightening treatment such as Sublime to lift up the corners of the mouth is what I would recommend. I would place the filler in the cheeks, nasolabial folds, corners of mouth, marionette lines and prejowl hollows. 

Those mounds of fat you're referring to occur in every single individual as facial aging occurs as a result of volume loss. The best way to bring them up is to have a procedure called autologous fat grafting which will restore the volume back to where it once used to be. This procedure will result in making you look twenty years younger while looking just like yourself at a younger age and the results are nearly permanent. Good luck from New York City and find yourself a plastic surgeon who is well-versed in autologous fat grafting techniques