Sadly not Henry’s first time for the cooler (8)
I believe the answer is:
monteith
'the cooler' is the definition.
Both the definition and answer are singular nouns.
Perhaps you can see a link between them that I don't see?
'sadly not henry's first time' is the wordplay.
'sadly' is an anagram indicator.
'first' indicates taking the first letters.
The initial letter of 'henrys' is 'h'.
'not'+'h'+'time'='nothtime'
'nothtime' anagrammed gives 'MONTEITH'.
'for' acts as a link.
(Other definitions for monteith that I've seen before include "Handkerchief" , "Silver bowl" .)