Flyer allowed to cover old Republican president (8)
I believe the answer is:
lorikeet
'flyer' is the definition.
(type of bird)
'allowed to cover old republican president' is the wordplay.
'allowed to' becomes 'let' (to let something happen is to allow it).
'cover' is an insertion indicator.
'old' becomes 'o' (common abbreviation eg in OE for Old English).
'republican' becomes 'r' (abbreviation).
'president' becomes 'ike' (nickname of Dwight Eisenhower).
'o'+'r'+'ike'='orike'
'let' enclosing 'orike' is 'LORIKEET'.
(Other definitions for lorikeet that I've seen before include "Small parrot" , "Kite role (anag) - parrot" , "Bird -- any of the small lories" .)