I had never even heard of Mini-DVD before reading your post. So, I googled it. Here are two suggestions for you after reading an article online:

1) Are you sure the mini-DVD disc from the camcorder is finalized? You may have to put it back into the camcorder to check this.

2) Can your computer read the mini-DVD disc? If so, you may be able to transfer the material onto your computer and then burn a regular DVD-R.

It may be that your Sony DVD player just can't read the mini-DVD discs. Unfortunatley, format incompatibility happens. Hope this helps..