The most common and most likely answer to this crossword clue is the 11 letter word IMPORTUNATE
I've seen this crossword clue in The Times
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More clues leading to the result IMPORTUNATE
Buddy explains!
The straight part
We know that "Persistent" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case persistent = importunate.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 IMPORTUNATE mean?
IMPORTUNATE adjective- (of a demand) Persistent or pressing, often annoyingly so.
- (of a person) Given to importunate demands, greedily or thoughtlessly demanding.
- To importune, or to obtain by importunity.