The most common and most likely answer to this crossword clue is the 5 letter word FALSE
I've seen this crossword clue in The Times
F
A
L
S
E
More clues leading to the result FALSE
- Some tales La Fontaine rejected could be untrue
- Note London School of Economics is wrong
- Sports body going to college in London? Not so
- Fake news: pleas FA regularly rejected
- Phoney female heading a London school
- Unfaithful wife finds loose-ish part of support structure
- Organisation representing soccer school in London? Wrong
- Iron also reduced inside? Not true
- Mock exams seeming laughable at first, initially upset
- Logical result of Cornish river and home counties being adjacent?
Buddy explains!

The straight part
We know that "Fake" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case fake = false.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 FALSE mean?
FALSE noun- One of two options on a true-or-false test.
- Untrue, not factual, factually incorrect.
- Based on factually incorrect premises.
- Spurious, artificial.
- Of a state in Boolean logic that indicates a negative result.
- Uttering falsehood; dishonest or deceitful.
- Not faithful or loyal, as to obligations, allegiance, vows, etc.; untrue; treacherous.
- Not well founded; not firm or trustworthy; erroneous.
- Not essential or permanent, as parts of a structure which are temporary or supplemental.
- Used in the vernacular name of a species (or group of species) together with the name of another species to which it is similar in appearance.
- Out of tune.
- In a dishonest and disloyal way; falsely.