Hi there sorry to hear about all you've gone through - Christine is right about the loss of swallowing reflex during radiation, I'm surprised they performed surgery after the rads. Radiation also causes complications with healing... Most tonsil related cancer is HPV+. Often they will remove the tonsil tissue then do radiation and chemo. Has she been scanned? I'm assuming so - she should have another scan to see if it's anywhere else - though this type of cancer usually responds well to radiation, I'm wondering if the stuff by the carotid wasn't there from the outset... Did they radiate her neck? People can be re radiated but it can cause a lot of side effects or worsen the ones she already has. There are head and neck exercises that can be done to try and help stretch the muscles in her mouth and throat.
Christine is right about the nutrition, is she able to do fluids at all if so it may clear sime of the mucous... Also with regards to the mucous - i found a waterpik helped clear out the gunk, i used to add bit of baking soda - alcohol free mouthwash and shoot it aroung my mouth - on the lowest setting. It made me feel fresher. I still use it. Some people here may poo poo this suggestion but I have found acupuncture has helped with the swallowing, there are studies to support this but you should definitely see a therapist to do a swallow study and see why - but first an foremost do go to a ccc if at all possible - also try adding some protein powder to her formula - just make sure it's not clumpy- dilute it if you have to - it will help with healing, and add more calories - take care - I'm sure others here will be along to suggest...