The most common and most likely answer to this crossword clue is the 8 letter word PRESENCE
I've seen this crossword clue in Independent
P
R
E
S
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N
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E
More clues leading to the result PRESENCE
- Spirit of quiet home this writer had abandoned
- Being There, for which acting needs time given over to Sellers’ lead
- Shortly attending church for spirit
- Republican VP hedging about the end of his political influence
- Company submits reports
- 9 gifts for company
- Impressive carriage in attendance?
- Magnetism some research discovered in coins
- Demeanour with which gifts may be announced
- Aura of change embracing sleeveless dress
Buddy explains!

The straight part
We know that "political influence" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case political influence = presence.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 PRESENCE mean?
PRESENCE noun- The fact or condition of being present, or of being within sight or call, or at hand.
- The part of space within one's immediate vicinity.
- A quality of poise and effectiveness that enables a performer to achieve a close relationship with their audience.
- A quality that sets an individual out from others; a quality that makes them noticed and/or admired even if they are not speaking or performing.
- Something (as a spirit) felt or believed to be present.
- A company's business activity in a particular market.
- An assembly of great persons.
- The state of being closely focused on the here and now, not distracted by irrelevant thoughts.
- To make or become present.
- (audio) The ambient sounds in a room or on a set.