Immaturity of horse carrying on excitedly at first (6)
I believe the answer is:
nonage
'immaturity' is the definition.
(I've seen this before)
'horse carrying on excitedly at first' is the wordplay.
'horse' becomes 'nag' (term for an old horse).
'carrying' indicates putting letters inside.
'at first' indicates taking the first letters.
The initial letter of 'excitedly' is 'e'.
'nag' placed around 'on' is 'nonag'.
'nonag'+'e'='NONAGE'
'of' is the link.
(Other definitions for nonage that I've seen before include "Years before adulthood" , "State of being unable to marry" , "In law, state of being under age for certain things" , "very early life" , "Period of youth before maturity" .)