Ship carrying ex-PM over Thames, perhaps (9)

I believe the answer is:

mayflower

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'ship' is the definition.
(I have seen 'old ship' mean 'mayflower' so perhaps 'ship' could also mean 'mayflower')

'ex-pm over thames perhaps' is the wordplay.
'ex-pm' becomes 'May' (Theresa May former prime minister).
'over' says to put letters next to each other.
'thames perhaps' becomes 'flower' (I am not sure about this - if you are sure you should believe this answer much more).
'may'+'flower'='MAYFLOWER'

'carrying' is the link.

Can you help me to learn more?

(Other definitions for mayflower that I've seen before include "Pilgrims Fathers' ship" , "traditional craft" , "Pilgrim Fathers' ship to the US" , "emigrants' vessel?" , "Pilgrim Fathers' vessel" .)

I've seen this clue in The Hindu.

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