What causes such shadowing around the mouth and how would you treat it? (Photo)

Thank you for your question! As we age, our fat pads start to migrate downward and separate. We also have bone loss. With the natural facial aging process, we can start to develop shadowing in areas around the mouth. This area can be treated by replacing volume loss and creating structure to support the fat pads. A hyaluronic-based dermal filler treatment (like Restylane or Vollure for example) is ideal for volume replacement in this area. In contrast, neuromodulators (like Botox)relax the muscle and are excellent for softening fine lines. In certain muscles, neuromodulators can help upturn the corners of the mouth as well. We recommend consulting with an experienced injector to create a treatment plan that will best address your needs. Good luck to you on your aesthetic journey!

Your question is difficult to answer without pictures, but it seems that the Botox was incorrectly placed and is now affecting the muscles involved with your smile. It can take up to 2 weeks to see the full results from Botox, but your smile likely won't get better until the Botox wears off in 3-4 months. I would recommend you follow up with your injecting physician. If you don't feel comfortable with him/her then find someone you feel more comfortable with. 

Age and gravity causes the soft tissue of the mid and lower face to fall, causing shadowing around the mouth and chin. Both medium-viscosity fillers such as Vollure® and Restylane® Defyne and/or Ultherapy® can lift this sagging tissue and improve this problem. 

The appearance of shadowing around your mouth is the result of the contours and the structure of your face in that area.  It is not the result of muscle animation at all; in fact, it appears from the image you've provided that your face is completely in repose (at rest), and the contours are still there.  Thus, any botulinum toxin treatment would at the very least not help at all, and at worst would distort your normal facial functions, like smiling, eating, drinking, etc, and make things a whole lot worse.  I definitely advise you to proceed with caution with Botox around the mouth area.
What then should you do?  While you haven't given me your age, and I haven't had the opportunity to examine you in person and get a detailed assessment of what your facial structure is like, I think I can gather enough information from just this image to give you a couple of good options.  What is really going on here is that you have a scarcity of subcutaneous tissue thickness and volume in key areas around your mouth, whether due to the aging process, some medical issue like extreme weight loss, or just genetics.  That part really isn't as important, as for me the solution is pretty much the same:  add volume strategically in the areas of the face that need it to soften and camouflage those contours.  This will produce a fresher, less harsh, more youthful, and pleasant appearance.
You can use a variety of fillers off the shelf to accomplish this, including HA (many like Restylane, Juvederm, Versa, and probably many others in Europe we can't get here in the US), PLLA (Sculptra), or CHA (calcium hydroxyapatite - Radiesse).  Most of these fillers will do pretty much the same thing, with some variations in longevity of results, risks depending upon who is doing it and how they do it, and efficacy.  The big advantage to fillers is the ease of use, economical cost, and safety.
In my opinion, though, the best option for you to consider is autologous fat transfer from somewhere else on your body, like your abdomen or thighs, to your face.  I have done this on many younger patients who desired volume correction in areas of deficiency due strictly to genetics and facial structure.  The great thing about fat transfer is that in addition to the obvious volume correction, there is also a nutritive effect from the other cells, like stem cells, that are in the transferred adipose tissue.  These cells produce a variety of growth factors which stimulate things like increased blood supply in the tissues, increased collagen production, and a softening of fibrous or scarred tissues.  These things don't happen with fillers.  Additionally, because it is really your own tissue, if done properly, the risks are very minimal, and the results will be very long lasting, even "permanent" to the degree that anything we do can be permanent.  The main thing about fat transfer is that it is imperative that it be done by someone with the appropriate knowledge, training, and skill.  It is very dependent upon good technique and the use of the proper instruments and equipment, thus you should really research the doctor doing this if you are going to go that route.  Best of luck!