What about TWOK? It gave Kirk a never-before-mentioned adult son, along with the lost love who was his mother
Even from the first time I saw TWOK I figured Carol Marcus was the "little blonde lab technician" Gary Mitchell "aimed" at Kirk. While lying in sickbay in "Where No Man Has Gone Before" Mitchell boasts that he outlined her whole campaign for her. Kirk sputters, "I almost married her!"
Why would Kirk almost
have married her—because he got her pregnant, perhaps? There's a lot of exposé in WNMHGB. Kirk was very serious at the academy, "a stack of books with legs." We also know that Kirk and Mitchell were indestructible, young action heroes sniffing out trouble wherever they could find it. Mitchell reminds Kirk of a poisoned dart he took for Kirk on Dimorus.
This dovetails with the quarrel between Kirk and Carol Marcus on Regula. Kirk knew about David ("Is that David?"). And we know why Kirk almost-but-didn't marry the blonde lab technician—she knew Kirk is a danger junkie and didn't want to lose her son, temporarily or permanently, to the same dangerous, never-at-home lifestyle.
I'd say there's abundant foundation in TOS for an adult son to appear in TWOK. Add however many years Kirk had been out of the academy in TOS to the fifteen years since he'd last seen Khan, and David is no surprise. That Carol is a scientist is also grounded in WNMHGB.