Giving the reins when the horse drops its head isn't much of a reward. Suddenly the contact is gone and the horse throws its head.
A horse should be very established in the contact before you start giving away the reins. I got very, VERY told off by my coach, who rides at an international level, for giving away my inside rein by less then 2cm, to scratch a green horse's neck when he came through. I got told off, because at this stage in the horse's training, it is not yet balanced and confident enough in the contact yet, to support itself when you give a rein away and change your balance.
Not sure if this was in response to my post, but "giving" or "offering" as I'm used to hearing it and "completely abandoning contact" (what you seem to be describing) are markedly different. From my own experience I can say that developing the level of contact that is truly correct, stable and distributed evenly has taken a while to learn and changes as you grow as a rider and team. furthermore, and if that wasnt hard enough, it is different for every horse, making it even MORE elusive for someone learning.
What I think tends to happen in an honest effort to master this (especially with no trainer from the ground and without a schoolmaster) is that the rider will brace against the horse, horse will brace against rider and tense up, which really gets nothing done. Some horses will even take this opportunity to lean against the riders hands and freight train onto the forehand. So explaining and learning to follow, with softness and offer self carriage is is just as if not more important than asking.
Just a thought, I have no idea what this pair's issues are! (that's not a cut, we all haven 'em, and anyone who says they don't is a liar.)
OP all of this stuff is hard to learn, too and at the end of the day, no post on the Internet or page in a book will let you feel what you need to feel and that's where you really learn. Real time explanations from your trainer will be the best. (I realize he/she is not around).