Admirer's initial longing to construct a letter (5)
I believe the answer is:
aitch
'a letter' is the definition.
(aitch is a kind of letter)
'admirer's initial longing' is the wordplay.
'initial' indicates taking the first letters.
'longing' becomes 'itch' (I've seen this before).
The initial letter of 'admirers' is 'a'.
'a'+'itch'='AITCH'
'to construct' acts as a link.
(Other definitions for aitch that I've seen before include "character leading heroes" , "bone" , "Letter sound that's often dropped?" , "8th letter of the alphabet" , "what some drop?" .)