Problem with head of shire horse (4)
I believe the answer is:
snag
'problem' is the definition.
(snag can mean a problem or difficulty)
'head of shire horse' is the wordplay.
'head of' suggests taking the first letters.
'horse' becomes 'nag' (term for an old horse).
The first letter of 'shire' is 's'.
's'+'nag'='SNAG'
'with' acts as a link.
(Other definitions for snag that I've seen before include "Temporary hold-up" , "Sharp protuberance; catch" , "Rip or tear" , "banger down under" , "Minor difficulty" .)