Request the presence of popular vet working across island (6)
I believe the answer is:
invite
'request the presence of' is the definition.
(I've seen this before)
'popular vet working across island' is the wordplay.
'popular' becomes 'in'.
'working' indicates anagramming the letters.
'across' means one lot of letters goes inside another.
'island' becomes 'i'.
'vet' is an anagram of 'vte'.
'vte' going around 'i' is 'vite'.
'in'+'vite'='INVITE'
(Other definitions for invite that I've seen before include "Ask round" , "Ask to one's home in friendly way" , "Ask for, say comments" , "Ask (someone to visit)" , "Request (someone to attend)" .)