What injectable should be used to raise the corners of my mouth? (photo)

Combining Hyaluronic Acid based fillers with Injecting the DOA muscle will definitely make a difference.

A downturned smile or permanent frown can be improved with a combination approach, including neuromodulators (such as Botox, Dysport, and Jeuveau), injectable fillers, and radiofrequency skin tightening treatments such as Thermage or Venus Legacy. I recommend getting a formal evaluation with an experienced cosmetic dermatologist to develop a treatment plan that works for you. Best, Dr. Emer.

The marionette lines as well as the downturned corners of mouth are commonly treated using hyaluronic acid based fillers like Restylane Refyne/Defyne. 

First and foremost one must treat the cause versus the symptom.  Many times the  naso labial folds and marionette lines are caused from the lack of volume in the upper face.  To improve these conditions I utilize Sculptra first and then I address the lower face with a hylauronic filler such as a Juvederm or Restylane product.  

Fillers such as Juvederm or Restylane is a very good option
to help correct the marionette lines. In addition, it can be placed at the
downward corners of the mouth to give an appearance of a lift.

It would have been helpful to see your entire face in your photo.  Volume loss occurs over the entire face and volume loss in mid face can make marionettes, as you have shown and described, worse because gravity exacerbates this area.  I often treat cheeks AND lower face to lift and support skin as well as create "triangle of youth".  If you consistently treat the lower face without addressing the midface, you may correct to lines you see, but your face shape will turn boxy which is not youthful or flattering.  I recommend Voluma or Sculptra for the cheeks, and Juvederm to lower face.  Hope this information is helpful!

Hyaluronic acid fillers (Juvederm, Restylane, Beletero, Teosyal, Revanesse) are a great choice to lift and support the lower face, along with Botox to the DAO muscles to create a slight upturn to the smile. I would suggest your practitioner evaluate and treat the whole face with fillers to create a refreshed look. Treating a single area on a mature face can lead to a imbalance in the features. Choose a reputable provider and ask them to present you with a treatment plan for a non-surgical facelift. Best, Dr. Tkatch

Most Restylane or Juvederm products will work well.  I would avoid the very light and heaviest ones. I would also avoid Radiesse and sculptra.  You can start with Juvederm ultra and adjust treatment after that if need be.  

In my opinion the single best way to rejuvenate the face in almost any age is with the use of autologous fat grafting. The reason the face droops as a person gets older is because just like a balloon it deflates. Re-inflating it with your own fat will restore the more youthful appearance and better toned skin. If you do not wish to have fat grafting performed, then the next choice would be to use a filler like Restylane injected into the marionette lines themselves and still combine it with Botox to treat the depressor anguli Oris muscle. I hope this helps and good luck from New York City