The most common and most likely answer to this crossword clue is the 10 letter word OLEAGINOUS
I've seen this crossword clue in Financial Times
O
L
E
A
G
I
N
O
U
S
Buddy explains!
The straight part
We know that "Oily/obsequious" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case oily/obsequious = oleaginous.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 OLEAGINOUS mean?
OLEAGINOUS adjective- Oily, greasy.
- (of manner or speech) Falsely or affectedly earnest; persuasively suave.