Looking embarrassed, volunteers given exercise in bureaucracy (3,4)
I believe the answer is:
red tape
'bureaucracy' is the definition.
(thesaurus)
'looking embarrassed volunteers given exercise' is the wordplay.
'looking embarrassed' becomes 'red' (I've seen this before).
'volunteers' becomes 'ta' (abbreviation for Territorial Army).
'given' says to put letters next to each other.
'exercise' becomes 'pe' (Physical Education).
'red'+'ta'+'pe'='RED-TAPE'
'in' is the link.
(Other definitions for red tape that I've seen before include "legal agreement?" , "(Obstructive) bureaucracy" , "Petty official procedures" , "unnecessary paperwork" , "European regulations?" .)