Letter has half of them worried in recession (5)
I believe the answer is:
theta
'letter' is the definition.
(theta is a kind of letter of the alphabet)
'half of them worried in recession' is the wordplay.
'half of' indicates taking half.
'worried' becomes 'ate' (I've seen this before).
'in recession' is a reversal indicator.
'ate' reversed gives 'eta'.
'them' cut in half is 'th'.
'th'+'eta'='THETA'
'has' is the link.
(Other definitions for theta that I've seen before include "8th letter of the Greek alphabet" , "character from Greece" , "Character from overseas" , "character in Euripides" , "character in Marathon" .)