The cheese that broke Herbie’s heart (4)
I believe the answer is:
brie
'the cheese' is the definition.
(Brie is an example)
'broke herbie's heart' is the wordplay.
'broke' indicates an anagram.
'heart' means to look at the middle letters.
The centre of 'herbie' is 'erbi'.
'erbi' anagrammed gives 'BRIE'.
'that' is the link.
(Other definitions for brie that I've seen before include "Soft and creamy mild white cheese" , "dairy product" , "Prov. squalid place" , "Variety of cheese" , "French cream cheese" .)