Crossword Genius

Tube pushed back after daughter prods device, I'm off! (6-3)

Ross

I believe the answer is:

toodle-pip

I'm a little stuck... Click here to teach me more about this clue!

'i'm off' is the definition.

'tube pushed back after daughter prods device' is the wordplay.
I cannot really see how this works, but
'daughter' could be 'd' (genealogical abbreviation) and 'd' is present in the answer.
'device' could be 'tool' (I've seen this before) and 'tool' is found within the remaining letters.
This explanation may well be incorrect...

Can you help me to learn more?

(Other definitions for toodle-pip that I've seen before include "Farewell remark - dipole top (anag)" , "Farewell (informal)" , "Goodbye (dated)" , "so long" , "Take care" .)

I've seen this clue in The Telegraph.
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