The most common and most likely answer to this crossword clue is the 8 letter word RHETORIC
I've seen this crossword clue in Financial Times
R
H
E
T
O
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C
More clues leading to the result RHETORIC
- Right-hand man keeping to hyperbole
- Rioter oddly cold inhaling hot, hot air
- Articles drawn from Cairo and Tehran read back in persuasive speech
- Playing concerto’s finale, Richter displays eloquence
- Turning hot and cold, rioter displaying verbal skill
- Grand words spoken as Gable’s iconic character or Bogart’s
- Right hand erotic manipulation: it’s often longwinded
- Eloquence of mutinous choirmasters missing Mass
- Relocate to richer address
- Not a hierocrat spouting bombast
Buddy explains!

The straight part
We know that "Eloquence" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case eloquence = rhetoric.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 RHETORIC mean?
RHETORIC noun- The art of using language, especially public speaking, as a means to persuade.
- Meaningless language with an exaggerated style intended to impress.
- Part of or similar to rhetoric, the use of language as a means to persuade.
- Not earnest, or presented only for the purpose of an argument.