One of the main reason for neck pain when swallowing is due to sore throat, infection in the throat or a simple toothache. Swollen lymph node likes tonsils, adenoids are also sometimes responsible for neck pain after eating.
Next Steps
some other causes are throat injury ,ear infection,cold and flue, cough,acid reflux consult your doctor
Health Tips
apply a heat pad or ice pack right after you notice the pain during the swallowing of the food is the good option