Stubborn cousin ate in a mess (9)
I believe the answer is:
tenacious
'stubborn' is the definition.
(similar in meaning)
'cousin ate in a mess' is the wordplay.
'in a mess' indicates anagramming the letters.
'cousin'+'ate'='cousinate'
'cousinate' with letters rearranged gives 'TENACIOUS'.
(Other definitions for tenacious that I've seen before include "Persistent in holding on" , "refusing to give up" , "Holding fast" , "Persevering" , "Determined" .)