Fabric produced by girl briefly engaged by company (6)
I believe the answer is:
calico
'fabric' is the definition.
(I know that calico is a type of fabric)
'by girl briefly engaged by company' is the wordplay.
'by' means one lot of letters go next to another.
'girl briefly' becomes 'Ali' (short for Alison).
'engaged by' becomes 'c' (I can't justify this - if you can you should believe this answer much more).
'company' becomes 'co' (abbreviation).
'ali' put after 'c' is 'cali'.
'cali'+'co'='CALICO'
'produced' is the link.
Can you help me to learn more?
(Other definitions for calico that I've seen before include "Plain cotton cloth, white or unbleached" , "Coarse printed cotton cloth" , "Type of cotton cloth" , "White cotton fabric" , "Given material for dress" .)