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?
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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.