The most common and most likely answer to this crossword clue is the 5 letter word THEIR
I've seen this crossword clue in The Guardian
T
H
E
I
R
More clues leading to the result THEIR
- Genderless term the Irish must abandon to some extent
- The third person’s one taking over after time
- Those people’s time with the next in line
- Start to tax one who receives the estate of those people
- Affluent heiresses, in reduced circumstances, becoming possessive
- One who inherits after time what’s belonging to them
- Article I at first rightly associated with others
- Other people's time with next of kin?
- That group’s taking time – one expected to succeed
- Possessive person who’ll succeed on time
Buddy explains!

The straight part
We know that "Possessive" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case possessive = their.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.