Intensify correspondence with letters from admirers (3,4)
I believe the answer is:
fan mail
'letters from admirers' is the definition.
(letters to a celebrity from their admirers)
'intensify correspondence' is the wordplay.
'intensify' becomes 'fan' (fanning is a kind of intensifying).
'correspondence' becomes 'mail' (mail is written correspondence).
'fan'+'mail'='FAN-MAIL'
'with' acts as a link.
(Other definitions for fan mail that I've seen before include "Admiring post" , "Letters to a celebrity" , "Appreciative letters" , "Letters sent to public figures from admirers" , "Letters from a supporter" .)