The most common and most likely answer to this crossword clue is the 9 plus 5 letter word PRESIDENT ELECT
I've seen this crossword clue in Financial Times
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More clues leading to the result PRESIDENT ELECT
Buddy explains!
The straight part
We know that "one (almost) in White House" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case one (almost) in white house = president elect.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 PRESIDENT ELECT mean?
PRESIDENT ELECT noun- (sum of parts) A person who has been elected to a presidency but has not yet been inducted into office.