Tired out with Tony's initial American rubbish (8)
I believe the answer is:
detritus
'rubbish' is the definition.
(detritus is a kind of rubbish)
'tired out with tony's initial american' is the wordplay.
'out with' indicates an anagram (out can mean wrong or inaccurate).
'initial' indicates taking the first letters.
'american' becomes 'us'.
The initial letter of 'tonys' is 't'.
'tired' mixed up with 't' can make 'detrit'.
'detrit'+'us'='DETRITUS'
(Other definitions for detritus that I've seen before include "waste material" , "Accumulated debris" , "Bits and pieces of rubbish" , "Worn down rocks, gravel etc." , "Left over bits and pieces" .)