The most common and most likely answer to this crossword clue is the 9 letter word EQUATIONS
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More clues leading to the result EQUATIONS
Buddy explains!

The straight part
We know that "relationships" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case relationships = equations.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does EQUATIONS mean?
EQUATIONS noun- The act or process of equating two or more things, or the state of those things being equal (that is, identical).
- An assertion that two expressions are equal, expressed by writing the two expressions separated by an equal sign; from which one is to determine a particular quantity.
- A small correction to observed values to remove the effects of systematic errors in an observation.