Do I always need a relative word after the category noun?

No. "The thing I noticed" and "the reason she called" are natural without one. "The person who helped" and "the place where we waited" keep the relative word in the patterns practised here.

What can follow "the way" in this exercise?

You can use no connector, "that", or the more formal "in which". Do not use "how" after "the way" in these answers.

Home Grammar Cleft Sentences ExercisesThing-Cleft Exercises

Thing-Cleft Exercises

B2-C1 Level

English Exercises is teacher-led by Charlie. About the teacher · How content is made

The name is a little misleading. These sentences can focus on a reason, place, person, thing, time or way. I'd check the opening noun first. It guides the next part: the person who, the place where or the time when.

With the thing, the connector that is often omitted. After the way, this exercise accepts no connector, that, or in which. Keep the original time reference so that is and was still fit the meaning.