A fish by itself derived nutrient from algae (4-4)
I believe the answer is:
agar-agar
'derived nutrient from algae' is the definition.
(agar-agar is derived from some species of algae)
'a fish by itself' is the wordplay.
'fish' becomes 'gar' (type of gar).
'by itself' shows that letters should be duplicated (the word appears by another version of itself).
'a'+'gar'='agar'
'agar' doubled is 'AGAR-AGAR'.
(Other definitions for agar-agar that I've seen before include "Gelatinous thickening for food" , "Jelly from seaweed" , "Thickener obtained from seaweed" , "Food thickener from seaweed" .)