If upset by the dog having no tail, it's still a pet name (4)
I believe the answer is:
fido
'pet name' is the definition.
(I've seen this in another clue)
'if upset by the dog having no tail' is the wordplay.
'upset by' indicates an anagram.
'having no tail' means to remove the last letter (I've seen 'has no tail' mean this).
'dog' with its final letter taken off is 'do'.
'if' anagrammed gives 'fi'.
'fi'+'do'='FIDO'
'it's still a' acts as a link.
This may not be right. Some or all of it may belong to another bit of the clue.
Can you help me to learn more?
(Other definitions for fido that I've seen before include "it would disperse fog" , "Typical dog's name" , "Fog disperser" , "Dog perhaps" .)