Crossword Genius

Publishing company with marketing men supporting single out on the radio (7)

Ross

I believe the answer is:

picador

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

'single out on the radio' is the definition.
Although both the answer and definition are singular nouns, I don't understand how one could define the other.

'publishing company with marketing men supporting' is the wordplay.
'publishing company with' becomes 'pic' (I can't justify this - if you can you should believe this answer much more).
'marketing' becomes 'ad' (advertisement is a type of marketing).
'men' becomes 'OR' (military abbreviation for Other Ranks).
'supporting' is a charade indicator (letters next to each other) (in a down clue, some letters go below others).
'pic'+'ad'+'or'='PICADOR'

Can you help me to learn more?

(Other definitions for picador that I've seen before include "Lancer" , "Horseman in bullring" , "fighter with lance" , "ringfighter" , "Riding bullfighter" .)

I've seen this clue in The Guardian.
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