Here is a precise statement. Suppose every set belongs to an inner model of a measurable Woodin (above X).
Suppose there is a club class of cardinals such that for all ,
Then there is an inner model of such that and such that is a class-generic extension of .
(Nothing special about GCH here). Though here one can require that is a fine-structure model if one wants.
The class can always be forced without adding sets.
If one has inner model theory for measurable Woodin cardinals then the class is unnecessary.