"How were they compacting the repaired core, especially at the top where it was steep? Were they wetting the clay while compacting?"
"They compacted with a skid steer and bucket. I did not see them wetting it."
A skid steer is not a compaction machine. Also, you cannot compact dry clay.
I suspect your leak is due to a little water passing through your core trench. I wonder if they were working with moist material down at the bottom and achieved sufficient compaction at that level? Was there a little water left in your old pond basin during the renovation?
I suspect your core trench could be re-excavated in the area of the suspected leak, plus some additional safety margin, and then sealed with a better compaction effort and your clay at the optimal moisture content. However, if your pond is not having a problem staying full, then probably not worth the cost.
Sorry to hear about the wet area. Before the era of concrete mega-dams, the old earthen dams used to be designed with a gravel toe drain to collect the water that would inevitably seep through the dam, which was then routed away.
Do you have some even lower ground beyond your soggy area? You may have to make a collection system (like a french drain) and run some corrugated flexible pipe to your lowest point to get rid of the water from a persistent dam leak.
