Press disrupted routine, holding Prime Minister up (9)
I believe the answer is:
importune
'press' is the definition.
(thesaurus)
'disrupted routine holding prime minister up' is the wordplay.
'disrupted' indicates an anagram.
'holding' means one lot of letters goes inside another.
'prime minister' becomes 'PM' (abbreviation).
'up' says the letters should be written in reverse (in down clue: letters go upwards).
'pm' written backwards gives 'mp'.
'routine' is an anagram of 'iortune'.
'iortune' placed around 'mp' is 'IMPORTUNE'.
(Other definitions for importune that I've seen before include "Solicit" , "Harass" , "badger" , "plague" , "press to give answers" .)