In the above phrase, the noun clause is "that he would spend up his inheritance."
This relative sentence is an appositive (that is a noun or pronoun, frequently with modifiers, placed next to another noun or pronoun to explain or distinguish it).
A noun clause can be defined as a dependent clause that replaces any noun in a sentence, whether it is a subject, an object, or a subject complement.
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Full Question:
What is the job of the noun clause in Brad's one worry that he would use up his inheritance never came true?