How can I fix the "foxing" of a shoe?
The most realistic fix is to buy a new pair of shoes...
If you must fix this somehow, be aware from the very first second that it will never look new or professional again and will probably break again very quickly.
First you need to cut away all the losee strands and pieces. The heel could probably be repaired with construction silicone (preferably white). I suggest applying several thin layers and rebuilding and sculpting the heel that way. You can shape the fresh silicone with your fingers if you dip them into a glass of water with lots of dish washing liquid.
The upper edge is pretty much a lost course. You could try hiding the damage by sewing on a new bias tape, but since some material is chewed away, it will always look bumpy. I would be very careful when sewing through so many layers. The needle bight break. Maybe glueing the bias tape on is the better solution.
That being said, whether you try to fix them or not, you'll probably end up replacing them very soon anyways.