I exported my vector CMYK file from Illustrator as an RGB image. On my customer's mobile, the colors change and he can not review the work to approve it.
What can I do to display the correct colors?
Educate your customer.
If they are hiring you to do some print work (evidenced by the CMYK setting), they should never review that work just casually on a phone. Ask them to take the time for the review and urge them to look at your comps on a computer.
As joojaa says in the comments, colour reproduction on a mobile device can be absolutely horrible, and is dependent upon a multitude of uncontrollable factors like mobile OS, viewing software, screen settings and ambient light.
Honestly, if a customer would review my print comps on a mobile, I'd be very insulted at how casually they take my hard work.