-The problem is,
thatproduction in her department has dropped.
thatwe didn't have these problems before she started working here.
It says that sometimes omitting the that creates a break in the flow of a sentence, a break that can be adequately bridged with the use of a comma.
Is it a rule or can we put 'that' after those commas?
You can have either the comma, or 'that'; you can't have both.