Implore disheartened intern to make a start (5)
I believe the answer is:
begin
'make a start' is the definition.
(I've seen this before)
'implore disheartened intern' is the wordplay.
'implore' becomes 'beg' (synonyms).
'disheartened' says to hollow out the word (remove centre letters) (the middle or heart is removed).
'intern' with its centre taken out is 'in'.
'beg'+'in'='BEGIN'
'to' is the link.
(Other definitions for begin that I've seen before include "Clandestine, furtive" , "Israeli prime minister and Nobel Peace Prize winner, d. 1992" , "Initiate" , "Launch" , "Found" .)