All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
There is no objection to the injection of local anesthetic to the mouth of a fasting person, etc. since the anesthetic is neither food nor drink. Also, it does not reach one's throat. However, one has to be careful of this medicine to avoid reaching the throat.
Allah knows best.