The most common and most likely answer to this crossword clue is the 13 letter word SOPHISTICATED
I've seen this crossword clue in The Daily Telegraph
S
O
P
H
I
S
T
I
C
A
T
E
D
More clues leading to the result SOPHISTICATED
- ‘Cosmopolitan‘ dictates I shop around
- Drops aitches it turned out, not right posh
- Ditches patois, turning suave
- ‘Cosmopolitan‘ dictates I shop after a fashion
- Worldly old Greek character on board drawn to defend criminal act
- Despot — his act, I suspect, is complex and efficient
- Stylish Sophia carrying short wand, on TED
- Doctor ditches patois for something cultured
- Smooth pistachio nuts eaten by horse? All but the middle
Buddy explains!

The straight part
We know that "posh" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case posh = sophisticated.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 SOPHISTICATED mean?
SOPHISTICATED verb- To make less natural or innocent.
- To practice sophistry; change the meaning of, or be vague about in order to mislead or deceive.
- To alter and make impure, as with the intention to deceive.
- To make more complex or refined.
- Having obtained worldly experience, and lacking naiveté; cosmopolitan.
- Elegant, refined.
- Complicated, especially of complex technology.
- Appealing to the tastes of an intellectual; cerebral.
- Dishonest or misleading.
- Impure; adulterated.