The most common and most likely answer to this crossword clue is the 5 plus 4 letter word SPARE TYRE
I've seen this crossword clue in Independent
S
P
A
R
E
T
Y
R
E
More clues leading to the result SPARE TYRE
- Rolling fat?
- Car insurance carried excess?
- Fat Alexander refused to do so after siege
- Duplicate Mediterranean city corporation
- A bit of extra weight put on ere pastry’s disposed of
- Thin exhaust, did you say? One’s at the back of car
- Thin buttery layer, essentially on fatty roll
- Don’t attack Lebanese city’s central expansion
- Extra fat needed after a blowout?
- Manifestation of corporate excess?
Buddy explains!

The straight part
We know that "Extra fat" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case extra fat = spare tyre.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.