The most common and most likely answer to this crossword clue is the 6 letter word EGOISM
I've seen this crossword clue in The Times
E
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O
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More clues leading to the result EGOISM
- Seignories regularly selected by male self-interest
- It concerns me if work is stopping me retiring
- Self-centredness, say, starts to offend in some measure
- . One ring with gems reset displays self-centredness
- Selfishness perhaps is in order
- Selfishness of horrible heartless moggies
- European travel is beginning to modify a selfish attitude
- Perhaps female shuns bravery for self-interest
- Self-seeking earl gets gismo repaired
- Only interest in me, say? I’m so extraordinary!
Buddy explains!

The straight part
We know that "Self-centredness" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case self-centredness = egoism.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 EGOISM mean?
EGOISM noun- The tendency to think selfishly with exclusive self-interest in mind.
- The belief that moral behavior should be directed toward one's self-interest only.
- (by confusion of the similar words) Egotism.