"after reading through" is correct and can be used in written English.
You can use it when you want to emphasize that you have carefully read through something, such as a article or report. For example: "After reading through the report, I noticed several inconsistencies."
Reading my mother's obituary, which he wrote, he stopped after reading through the list of his 12 surviving children and chuckled, "God, that's a lot of kids".
After reading through the front section, however, I found almost nothing on the race.
After reading through many of the entries, I realized soldier graffiti couldn't be foreign to Leipzig.
I watched it an hour or so after reading through the police report on Carlos's death.
After reading through a piece about infanticide called "The News Is Here," Cathay said, "Wow!
Mr. Washor said he found evidence of improper intubation after reading through hospital and ambulance call records.
After reading through the notes, the neurologist admonished me that nothing would be gained by pursuing this further if these spells happened so rarely.
After reading through Levine's book and reminding myself of my friends' faces and recollecting all I've learned about feeling successful in my own life, I signed the form, leaving both options blank.
Yes they are stupid and childish, but after reading through thousands of "normal" captions each week, they are sometimes the only things that can bring a smile to my cold, dead face.
But after reading through several dozen of the testimonies, amid the disquiet, I began to feel as if I were bearing witness to a small moment of human redemption.
"This doesn't sound like someone with a callous disregard for human life," said Imam Talib Abdur-Rashid, spiritual leader of the Mosque of Islamic Brotherhood in Manhattan, after reading through excerpts that dealt largely with the need to pray.
