read on
1. To read something on some location where information is posted or shared. I think I read something about that on the school's bulletin board. Don't believe everything you read on the internet, John.
2. To carry on reading. Sorry for the interruptions, class. Please read on, Tommy. Shall I read on, or do you want me to pause there for today?
read on
to continue to read. Please read on. Don't stop. She read on until she had come to the end of the story.