Funny to get new start delivering fish (5)

I believe the answer is:

tunny

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'fish' is the definition.
(I know that tunny is a type of food fish)

'funny to get new start' is the wordplay.
'funny' becomes 'tuny' (I am not sure about this - if you are sure you should believe this answer much more).
'to get' indicates putting letters inside.
'start' says to take the initial letters.
The first letter of 'new' is 'n'.
'tuny' enclosing 'n' is 'TUNNY'.

'delivering' acts as a link.

Can you help me to learn more?

(Other definitions for tunny that I've seen before include "Common food fish" , "Large marine fish" , "Warm water food fish" , "A tuna" , "Marine fish, the bluefin" .)

I've seen this clue in The Times.

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