Impressed by delivery date, turn to that man Jack to work (2,3,3)

I believe the answer is:

do the job

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'work' is the definition.
(thesaurus)

'impressed by delivery date turn to that man jack' is the wordplay.
'impressed by' is an insertion indicator.
'delivery date' becomes 'doob' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'turn' becomes 't' (abbreviation in Oxford Dictionary of Abbreviations).
'to that man' becomes 'he' ('he' refers to a man).
'jack' becomes 'J' (playing card abbreviation).
't'+'he'+'j'='thej'
'doob' enclosing 'thej' is 'DO THE JOB'.

'to' acts as a link.

Can you help me to learn more?

I've seen this clue in The Telegraph.

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