PM fine with second-class transportation to America once (9)

I believe the answer is:

mayflower

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'transportation to america once' is the definition.
(ship that took the Pilgrims to America in 1620)

'pm fine with second-class' is the wordplay.
'pm' becomes 'May' (Theresa May).
'fine' becomes 'f' (abbreviation).
'with' means one lot of letters go next to another.
'second class' becomes 'lower' (lower class is a kind of class. I am not sure about the 'second' bit.).
'may'+'f'+'lower'='MAYFLOWER'

Can you help me to learn more?

(Other definitions for mayflower that I've seen before include "bark used by settlers" , "Famous Pilgrim ship" , "emigrants' vessel?" , "Pilgrims' vessel" , "vessel crossing ocean" .)

I've seen this clue in The Telegraph.

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