One Republican woman, extremely tense, getting bug (8)
I believe the answer is:
irritate
'bug' is the definition.
(I know that bug is a more specific form of bedevil)
'one republican woman extremely tense' is the wordplay.
'one' becomes 'i' (Roman numeral).
'republican' becomes 'R' (abbreviation in US politics).
'woman' becomes 'rita' ().
'extremely' suggests removing the centre (only the extremes of the word are used).
'tense' with its middle taken out is 'te'.
'i'+'r'+'rita'+'te'='IRRITATE'
'getting' acts as a link.
(Other definitions for irritate that I've seen before include "badger" , "Make angry or fretful" , "bug" , "Wind up" , "Annoy, bother" .)