A short dance written by composer, primarily for cellist (6)
I believe the answer is:
casals
'cellist' is the definition.
(cellist Pablo Casals)
'a short dance written by composer primarily' is the wordplay.
'short dance' becomes 'sals' (I am not sure about this - if you are sure you should believe this answer much more).
'written by' means one lot of letters go next to another.
'primarily' suggests taking the first letters.
The first letter of 'composer' is 'c'.
'a'+'sals'='asals'
'asals' put after 'c' is 'CASALS'.
'for' is the link.
(Other definitions for casals that I've seen before include "Pablo (or Pau) -, cellist" , "Pablo -, noted Spanish cellist" , "Pablo --, famous cellist" , "Pablo --, Spanish cellist and composer 1876-1973" .)