... the leader of a town doesn't just have a kid... that'd be kinda weird... its possible he was illegitimate, but thats about it. If it was a secret birth, the town would wonder who his parents are.
But, even if he was Hokages son, he'd be about the same... more like Konohamaru I guess. Prolly less of a prankster.

BTW, why was he so against Ero-Sennins perveted nature, when he was sneaking into womens bathhouses in the 1st episode?