Yes, snoring can cause a sore throat after a nights sleep because of the air running through the airways in the person. Also, snoring means there is more sleeping problems, most likely sleep apnea. If waking up with a sore throat, chances are you snore throughout the night and it's causing irritation in your airways and in your throat. To help cure this, try seeing a doctor and asking them if there are any medications you can take to get rid of the snoring. Also, lose weight or try to stop drinking alcohol and the snoring may go away along with the sore throat.