Very loud newspaper is taking time to support rise of American advocate for voting rights (10)

'advocate for voting rights' is the definition.
(eg women's suffragists)

'very loud newspaper is taking time' is the wordplay.
'very loud' becomes 'suff' (I can't justify this - if you can you should believe this answer much more).
'newspaper' becomes 'rag' (informal term for a newspaper).
'taking' says to put letters next to each other.
'time' becomes 't' (abbreviation).
'suff'+'rag'+'is'+'t'='SUFFRAGIST'

'to support rise of american' acts as a link.
This may not be correct. Some or all of it may be part of another bit of the clue.