Welcome to the forum, Paul! I’m glad you posted - but I wish you’d joined back in spring so we could support you as you got started. There’s so much to learn and the clock starts ticking fast when you get your first colony 
Especially a nuc.
Feeding and supering are mutually exclusive, as a rule. If you have to feed, it’s to sustain or boost a weak or new colony or help a decent one survive hard times like winter months. So yes, putting a super on while feeding or even soon after is a no-go unless there is a sudden booming nectar flow and loads of workers to bring it in. Four to six weeks after your early May start would have been early to mid June, which is typically peak nectar flow in temperate regions like yours in Kansas and mine here in PA.
It’s also peak swarm season. So what I think happened is your healthy nucleus colony was right up to or at the point of preparing to swarm, and continued in that direction even though you gave them more space. Just a matter of bee timing/planning - usually very different from our desires or assumptions - that you will get the hang of with more learning and experience 
Colonies can swarm repeatedly, which results in dwindling honey stores and a critical shortage of workers, so nothing much gets done in the ‘parent’ hive. Sometimes a queen that’s raised to replace the one(s) that have left with the swarm(s) doesn’t get well mated or gets snatched by a bird and doesn’t return to the hive.
That’s the brood box (not base) in beekeeping terms 
Your pic was useful, thanks for posting it. I believe you’ll be able to put more photos in future posts now. I zoomed in on one definite and one possible queen cell on the frame you showed:
Did you notice this when you were inspecting - was it capped or open at the tip?
Bad sign of a pest incursion that will get worse with not enough workers. Without looking at it I don’t know if it’s beetle or wax moth larva. With no honey in your hive and very little or none to forage at this time of year your bees are probably starving, so not at their best in terms of patrolling the space.
Yes, remove the super asap. Whether the colony will make it is uncertain. Even if the queen cell is viable there won’t be time or enough drones around at this point for her to get mated before colder weather sets in. Your frame did have a lot of capped worker brood, but going on your mention of no (open) brood I doubt there’s an active queen present. You could see if you can buy a mated queen to introduce and possibly the colony will raise a decent amount of brood to boost the numbers and get through the winter. The workers being raised at this time of year need to be fatter than usual so they can help keep a survivable temperature while in cluster until spring. So if you want to give them a shot you’ll also need to feed, starting now and continuously until spring. I would recommend a combination of regular pollen patties and 2:1 simple syrup right now, to support a Hail Mary brood production. Then, you would switch to winter patties because they provide the carbs but with a lower protein content suited for winter survival:
https://www.betterbee.com/bee-feed/wp10-10lb-winter-patties.asp
This is also a critical time to get pests under control. There’s also the very good chance that your numbers have declined due to varroa mite infestation. None of the above will help them if you haven’t treated for mites or dealt with whatever laid the egg that produced that free ranging larva you saw.
I’d take everything I wrote as provisional, until you get a live experienced beek to come take a look with you. They can make the best assessment and guidance. Join a local beekeeping club and ask if someone can mentor you. They usually also offer beginner classes over the winter into early spring, so you can get more knowledge and some hands on practice under your belt before next season starts.
Wishing you good luck! And let us know how it goes from here 
