The most common and most likely answer to this crossword clue is the 7 letter word UNCTION
I've seen this crossword clue in The Times
U
N
C
T
I
O
N
More clues leading to the result UNCTION
- You’ve got oil on tunic, clumsy!
- Do without foot cream
- Jack ejected from meeting for insincere emotion
- Oily charm from judge leaving meeting
- Religious rite gets agreement out of court
- Court briefly, engrossed by wedding fervour
- Affected charm as court breaks up marriage
- Affected charm
- Being smarmy and dysfunctional to some extent
- Religious ceremony has to work but not loudly
Buddy explains!

The straight part
We know that "oil" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case oil = unction.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 UNCTION mean?
UNCTION noun- A salve or ointment.
- A religious or ceremonial anointing.
- A balm or something that soothes.
- A quality in language, address or delivery which expresses sober and fervent emotion.
- A smug, exaggerated use of language; smarminess.
- Divine or sanctifying grace.