Is this sentence correct? Especially the part that when..
Many Mothers can confirm
that when
their kids grow, they want to be mostly in the kitchen.
Yes, you can. It's fine. I suggest that you have a look at That-clause if you have the time.
In short, you used a noun clause (when their kids... in the kitchen
) which was an objective of the verb confirm. They are separate things.
You are parsing it wrong. The subordinate clause is "when their kids grow, they want to be mostly in the kitchen", subordinated by the word "that.
The clause "their kids grow" is subordinate to the first clause, subordinated by the word "when".
Perhaps rearranging it would make it clearer:
Many mothers can confirm that they want to be mostly in the kitchen when their kids grow.
If you don't want to rearrange or paraphrase your sentence, it's okay to just put another comma after "that," making "when their kids grow" parenthetical. Though you'll say it as if you just remembered it.