The most common and most likely answer to this crossword clue is the 11 letter word ADVENTURESS
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More clues leading to the result ADVENTURESS
Buddy explains!

The straight part
We know that "How about Becky Sharp" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case how about becky sharp = adventuress.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 ADVENTURESS mean?
ADVENTURESS noun- A female adventurer; a woman who seeks adventure.
- A woman of doubtful reputation or suspected character who pursues personal advancement, especially by sexual means; a demirep.
- A female varietist.