Yes, specific Youtube videos can be allowed/blocked for your child user by adding them to your white-list or blocklist (only if you are a Premium Mobicip user). For instance, if you wish to allow the video http://www.youtube.com/watch?v=waGECHA4IV0, just add 'waGECHA4IV0' to the white-list.
Mobicip Support May 2, 2016
Re: You Tube Videos blocked that I would like to allow