Swimmer egghead runs after too much (5)
I believe the answer is:
otter
'swimmer' is the definition.
(animal which swims)
'egghead runs after too much' is the wordplay.
'egghead' becomes 'e' (head letter of 'egg').
'runs' becomes 'r' (cricket abbreviation).
'after' means one lot of letters go next to another.
'too much' becomes 'OTT' (short for over the top).
'e'+'r'='er'
'er' put after 'ott' is 'OTTER'.
(Other definitions for otter that I've seen before include "one that eats fish" , "Something furry" , "Aquatic creature" , "Fish-eating member of weasel family" , "Good swimmer" .)