Run after Bach's first organ (7)
I believe the answer is:
bladder
'organ' is the definition.
(I've seen this before)
'run after bach's first' is the wordplay.
'run' becomes 'ladder' (both can mean a rip in tights).
'after' says to put letters next to each other.
'bach's first' becomes 'b' (1st letter of 'bach').
'ladder' put after 'b' is 'BLADDER'.
(Other definitions for bladder that I've seen before include "Chamber" , "organ" , "may be subject to inflation?" , "Bag in the lower abdomen" , "Bag containing liquid or gas" .)