Give the head of department a little publicity (10)

I believe the answer is:

administer

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'give the head' is the definition.
'administer' can be an answer for 'give' (administering is a kind of giving). I'm not certain of the 'the head' bit.

'of department a little publicity' is the wordplay.
'of' indicates an anagram (the answer comes from or 'of' the letters).
'department' becomes 'd' (abbreviation eg in MOD Ministry of Defence).
'little publicity' becomes 'minister' (I can't explain this - if you can you should believe this answer much more).
'd'+'a'='da'
'da' is an anagram of 'ad'.
'ad'+'minister'='ADMINISTER'

Can you help me to learn more?

(Other definitions for administer that I've seen before include "Dispense" , "Run" , "conduct" , "Manage" , "give" .)

I've seen this clue in The Globe and Mail.

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