Can children get gonnarhea from drinking after someone who has it

No, but there is plenty of mis-information out there, because there is so much gonorrhea of the throat in addition to the genitals, and so few people who realize that an infected penis can transmit it to the throat as well as the vagina and from vagina to penis, but not from the vagina to the mouth, or from mouth to mouth.
Here is one site that tries to explain it clearly: chlamydia - Forum on Safe Sex and HIV Prevention -- TheBody.com

That doctor says gonorrhea of the throat is rare, when actually there is a huge increase among young people in recent years, because they don't use condoms for oral sex and think they are safe.

Good information, but it is really not a good idea to drink after another person for other health reasons