Trade union repeatedly provides cover for routine performer (4)
I believe the answer is:
tutu
'cover for routine performer' is the definition.
(item worn by someone doing a dance routine)
'trade union repeatedly' is the wordplay.
'trade union repeatedly' can mean two replacements for 'trade union'.
'trade union' becomes 'TU' (abbreviation for trade union).
'trade union' becomes 'TU' (abbreviation for trade union).
'tu'+'tu'='TUTU'
'provides' acts as a link.
(Other definitions for tutu that I've seen before include "North African nation" , "Ballet dress" , "Female ballet costume" , "very short garment" , "In which dancer may appear" .)