If you’ve actually cut the clips you can turn them into subclips (cmmd/ctrl U) then highlight all those in the Project Panel and export them all at once.
Or
Take the long clip into Prelude and just mark all the sections you want as subclips then export those the same way. No cutting required. The benefit of this is that the master clip will retain that metadata so no when you load that big clip you’ll be able to easily have access to everything you’ve already marked.