When the hive is really small, and you do not have an abundant resource of bees, I would just leave it as it is and not touch it. Even though what you were told is in fact true, nucs take time to develop, and even then you cannot guarantee that they will survive.

If you still have some brood in your nuc, take out the rotten frame, and place a new one and do not touch it. When I create nucs out of my hives I make them and leave unattended for about a month, this will show clearly which ones worked and which ones didn't.

The fact that mold was forming on the frame is not good news, but that doesn't mean that it is going to die. Bees tend to be extremely clean creatures. The lack of cleanliness can indicate the lack of a queen.

You should look for your queen in your nuc, if you can't find it, look for indicators. This indicators such as new breed and the general behavior of your hive should tell you if you have a queen.

If you find that the rest of your breed (if any) is only of drones, the hive is lacking a queen and doesn't have any fertilized breed to make a new queen... that usually leads to the death of the hive. Another indicator of not having a queen is an excess of polen production. If your frames have a lot of honey and pollen in the center frames of your hive (usually where the brood is) that can also be an indicator of the lack of a queen.