One with royals next to one art gallery getting bug (8)
I believe the answer is:
irritate
'bug' is the definition.
(I know that bug is a more specific form of the action bedevil)
'one with royals next to one art gallery' is the wordplay.
'one' becomes 'i' (Roman numeral).
'with' is a charade indicator (letters next to each other).
'royals next' becomes 'irr' (I can't justify this - if you can you should believe this answer much more).
'to one art gallery' becomes 'tate' (Tate Modern, Tate Britain etc).
'i' put after 'irr' is 'irri'.
'irri'+'tate'='IRRITATE'
'getting' acts as a link.
Can you help me to learn more?
(Other definitions for irritate that I've seen before include "Chafe, annoy" , "Nark" , "Needle" , "Wind up" , "badger" .)