The game's up, shelving one fine romance with firm's Head of Production (1,4,3)
I believe the answer is:
a fair cop
'the game's up' is the definition.
'one fine romance with firm's head of production' is the wordplay.
'one' becomes 'a'.
'fine romance with' becomes 'fair' (I can't justify this - if you can you should give a lot more credence to this answer).
'firm' becomes 'co' (a firm is a company).
'head of' indicates taking the first letters.
The initial letter of 'production' is 'p'.
'a'+'fair'+'co'+'p'='A FAIR COP'
'shelving' acts as a link.
This may not be correct. It may be part of another bit of the clue.
Can you help me to learn more?
(Another definition for a fair cop that I've seen is " Deserved arrest or discovery as by an equitable policeman".)