Poor ruler given better brain supplement (8,3)
I believe the answer is:
thinking cap
'brain supplement' is the definition.
('put your thinking cap on')
'poor ruler given better' is the wordplay.
I cannot quite see how this works, but
'poor' could be 'thin' (I've seen this in other clues) and 'thin' is found in the answer.
'ruler' could be 'king' (king is a kind of ruler) and 'king' is found within the answer.
'better' could be 'cap' (both can mean to beat somebody else's attempt) and 'cap' is located in the answer.
No letters remain.
This explanation may well be incorrect...
Can you help me to learn more?
(Other definitions for thinking cap that I've seen before include "state of concentration" , "What should help one meditate" , "Supposed aid to cogitation" .)