The most common and most likely answer to this crossword clue is the 7 plus 6 letter word FORTUNE TELLER
I've seen this crossword clue in The Daily Telegraph
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More clues leading to the result FORTUNE TELLER
Buddy explains!

The straight part
We know that "Palmist" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case palmist = fortune teller.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 FORTUNE TELLER mean?
FORTUNE TELLER noun- A person who professes to predict the future in return for money.
- A cootie catcher.