Finish off project number one for Bronski Beat (5)
I believe the answer is:
throb
'beat' is the definition.
(I know that throb is a type of beat)
'finish off project number one for' is the wordplay.
'finish off' means to remove the last letter.
'project' becomes 'throw' (synonyms).
'number one for' becomes 'b' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'throw' with its final letter taken off is 'thro'.
'thro'+'b'='THROB'
'bronski' acts as a link.
I am not very happy about this link. It may belong to another bit of the clue.
Can you help me to learn more?
(Other definitions for throb that I've seen before include "Have strong or irregular heartbeat" , "Beat strongly" , "Drum" , "Vibrate with rhythm like heart" , "Strong rhythmic beat" .)