You can't get rid of the fat pads without surgery. However you can hide them by putting some filler into the tear trough just below the prolapsed fat. I would recommend either restylane of belotero for this. when you fill in the trough you will have a smoother transition from the cheek into the lower eyelid so that you won't see the shadows as much.
Hyaluronic acid is very good for this.
In particular for periocular treatment I find Restylane the best of the filler products for this type of area. It can be used to fill the scar and the under circle. The treatment is not permanent but generally it can provide approximately about of year of benefit. The key is finding a very experienced injector to do your treatment. This is typically going to be a physician or surgeon in one of the 4 core aesthetic specialties: dermatology, facial plastic surgeon, plastic surgeon, or oculoplastic surgeon. The best injectors do their own service and do not delegate to the nurses.