it is possible but any injury would likely be very minor (and accidental) since the throat is really not touched. a tongue depressor is used to push down the tongue in order to see the throat. the child may now have an unrelated sore throat gagging from post-nasal drainage or have a croupy voice from laryngeal irritation. things that have nothing to do with the examination.
can someone damage a small childs throat in any way by sticking a tongue depresser in too far? . she gagged until she vomited and now does not want to eat or drink and now sounds croopy.