Slur essentially influential local party suppresses (8)
I believe the answer is:
innuendo
'slur' is the definition.
(I've seen this before)
'essentially influential local party suppresses' is the wordplay.
I cannot quite see how this works, but
'local' could be 'inn' (I've seen this before) and 'inn' is found within the answer.
'party' could be 'do' (do is a kind of party) and 'do' is present in the answer.
This explanation may well be incorrect...
Can you help me to learn more?
(Other definitions for innuendo that I've seen before include "Disparaging oblique remark" , "Naughty suggestion" , "An unpleasant insinuation or intimation" , "Indirect implication" , "Indirect remark" .)