Without enthusiasm for a new diet that includes apple core (5)
I believe the answer is:
tepid
'without enthusiasm' is the definition.
(I've seen this before)
'a new diet that includes apple core' is the wordplay.
'a new' indicates anagramming the letters.
'that includes' is an insertion indicator.
'apple core' becomes 'p' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'diet' anagrammed gives 'teid'.
'teid' going around 'p' is 'TEPID'.
'for' is the link.
Can you help me to learn more?
(Other definitions for tepid that I've seen before include "Half-hearted" , "Unenthusiastic - cool" , "Showing little interest or enthusiasm" , "Slightly warm" , "Moderately warm" .)