Cheeky brat in hospital department, inspiring regret regularly (11)
I believe the answer is:
impertinent
'cheeky' is the definition.
(I've seen this before)
'brat in hospital department inspiring regret regularly' is the wordplay.
'brat' becomes 'imp' (I've seen this in another clue).
'hospital department' becomes 'ENT' (abbreviation for Ear, Nose and Throat department).
'inspiring' means one lot of letters goes inside another.
'regularly' means one should take alternating letters (regularly take one letter, leave next etc.).
The alternating letters of 'regret' are 'ert'.
'imp'+'in'+'ent'='impinent'
'impinent' going around 'ert' is 'IMPERTINENT'.
(Other definitions for impertinent that I've seen before include "of no relevance" , "Bold" , "brass-necked" , "Cheeky" , "not suitable" .)