They're reducing the light shed by city politicians on underworld figure (4,6)

'they're reducing the light shed' is the definition.
The definition suggests an adverb but the answer is not.

'by city politicians on underworld figure' is the wordplay.
'by' says to put letters next to each other.
'city' becomes 'LA' (Los Angeles).
'politicians' becomes 'MPs' (Members of Parliament).
'on' is a charade indicator (letters next to each other) (in a down clue, letters appear on others).
'underworld figure' becomes 'hades' (I can't explain this - if you can you should give a lot more credence to this answer).
'la' put next to 'mps' is 'lamps'.
'lamps'+'hades'='LAMP SHADES'

Can you help me to learn more?