One having links mostly with Republican president (5)
I believe the answer is:
chair
'president' is the definition.
(I know that president can be written as chair)
'one having links mostly with republican' is the wordplay.
'one having links' becomes 'chain' (a chain is made up of links).
'mostly' means to remove the last letter.
'with' says to put letters next to each other.
'republican' becomes 'r' (abbreviation).
'chain' with its final letter taken off is 'chai'.
'chai'+'r'='CHAIR'
(Other definitions for chair that I've seen before include "Movable seat" , "A committee's most important seat" , "director at meeting" , "academic office" , "Backed seat for one" .)