the "kan jeg" are merged together and I can't hear the "jeg" part, is it just me?
You are right! It sounds like "ka-eg" to me, haha
Actually, the correct word for this type of transaction in english is 'cheque'. Strange it should be marked wrong. Please get your house in order.
Can applies for physical requests. The correct translation is "MAY I pay by check"
While you are strictly correct correct, in modern British usage you would use 'can' for both ability and permission to do something. You would use 'may' if you wanted to be more polite