It may have its local head (4)
I believe the answer is:
boar
This might be a double definition.
'it may have its' is the first definition.
The definition suggests a singular noun which matches the answer.
'local head' is the second definition.
The definition and answer can be both animals as well as being singular nouns.
Perhaps you can see a link between them that I can't see?
Can you help me to learn more?
(Other definitions for boar that I've seen before include "Wild pig sounds tedious" , "Wild animal, sounds like drill" , "A male pig, sounds tedious" , "Resting place for birds or bats" , "The head for drink Falstaff had" .)