Investigate the cash-desk (5,3)
I believe the answer is:
check out
'investigate' is the definition.
(to check something out is to investigate it)
'the cash-desk' is the wordplay.
I cannot quite see how this works, but
'the' could be 't' (the is pronounced as a 't' sound in some dialects) and 't' is found within the answer.
'cash' could be 'c' and 'c' is located in the answer.
This may be the basis of the clue (or it may be nonsense).
Can you help me to learn more?
(Another definition for check out that I've seen is " Investigate - supermarket pay-desk".)