The that-clause is a "content clause" that functions as a "complement clause" of the noun phrase by "postmodifying" it.
In the sentence,
It was an admission that all might not be right between them.
the part [ an admission that all might not be right between them ] is a "noun phrase". In this noun phrase, [ an admission ] is postmodified with the that-clause: [ that all might not be right between them ].
This that-clause is not a relative clause. It is a content clause. To be more specific, it is a declarative content clause. According to Wikipedia, a content clause is a subordinate clause that provides content implied, or commented upon, by its main clause.
It is important to know the difference between the two constructions (relative clause and content clause). There are two important reasons.
First, they are different constructions. In the case of a relative clause, "one of its argument shares a referent with a main clause element on which the subordinate clause is grammatically dependent." (Wikipedia)
In the case of a content clause, its function is to complement, and it has no gap.
For instance, in "the person that saw you," the subject of the clause "saw you" is missing, but is understood to be "the person" that the NP as a whole refers to. Complement clauses do not usually have such a gap.
From What is a complement clause? (www.sil.org)
Second, in order to be postmodified by a that-clause, the noun must license it. We cannot do this with any nouns. A noun that licenses a content clause as its complement is usually about a thought, an opinion, a belief, or information of some kind.
Here is the related part on the page that-clause by Cambridge Dictionaries Online:
Noun + that-clause
We use a noun + that-clause to express opinions and feelings, often about certainty and possibility. We also use that with reporting nouns. Some nouns commonly used in this way are belief, fact, hope, idea, possibility, suggestion, statement, claim, comment, argument:
He is also having intensive treatment in the hope that he will be able to train on Friday.
Dutch police are investigating the possibility that a bomb was planted on the jet.