Cook coq au vin or duck, say, to go in cooler (2,4)

'go in cooler' is the definition.
The definition and answer can be both related to competing as well as being verbs in their base form.
Maybe they are linked in a way I don't understand?

'cook coq au vin or duck say' is the wordplay.
I cannot quite understand how this works, but
'cook' could be 'do' (do can mean to cook) and 'do' is located in the answer.
'duck say' could be 'bird' (duck is a kind of bird) and 'bird' is present in the answer.
This accounts for all the letters.
This may be the basis of the clue (or it may be nonsense).

'to' acts as a link.

Can you help me to learn more?