Nothing taking egghead to the river (4)
I believe the answer is:
nile
'the river' is the definition.
(Nile is an example)
'nothing taking egghead' is the wordplay.
'nothing' becomes 'nil' (eg nil is zero in football).
'taking' means one lot of letters go next to another.
'egghead' becomes 'e' (head letter of 'egg').
'nil'+'e'='NILE'
'to' is the link.
(Other definitions for nile that I've seen before include "Green" , "River of eastern Africa" , "Current" , "River of Egypt" , "World's longest river" .)