Dogged cousin ate out (9)
I believe the answer is:
tenacious
'dogged' is the definition.
(I know that dogged can be written as tenacious)
'cousin ate out' is the wordplay.
'out' indicates an anagram (out can mean wrong or inaccurate).
'cousin'+'ate'='cousinate'
'cousinate' is an anagram of 'TENACIOUS'.
(Other definitions for tenacious that I've seen before include "Persevering" , "Dogged" , "Holding fast" , "Keeping a firm hold" , "refusing to give up" .)