Peevishly incarcerating leader of Persians in Tyre (5-3)
I believe the answer is:
cross-ply
'peevishly incarcerating' is the definition.
I can't judge whether this definition defines the answer.
'leader of persians in tyre' is the wordplay.
'leader of' says to take the initial letters.
'in' is an insertion indicator.
'tyre' becomes 'crossly' (I can't explain this - if you can you should give a lot more credence to this answer).
The initial letter of 'persians' is 'p'.
'p' inserted into 'crossly' is 'CROSS-PLY'.
Can you help me to learn more?
(Another definition for cross-ply that I've seen is " Type of car tyre".)