I captured a small swarm of about 2 lbs of bees in the middle of June. I placed them in a NUC Box.
Over the past few weeks, hive moths have moved in since the swam is small. I tried removing all but 3 of the 5 frames from the Nuc box and cleaning out what moths I can see.
My question is, i’m pretty sure the bees are not going to make it in the nuc. How do I save the bee’s or is it a lost cause?
If you remove their queen, you could merge them with one of your other hives using a layer of newspaper. Nice boost to the other hive, and the bees get to live for a bit longer. 
As always, Thank you Dawn. I will let you know what happens.
If you decide to merge, you could put them in the same size box (8 or 10 frame) as your other hive, even if there are only 3 or 5 frames. They will be through the paper in 1-2 days, then you just take the extra box away. At this time of year, they probably won’t do anything crazy in the merging box if you only leave it on for 2 days. 
Hey @Dan_C sorry to hear about the moth infestation, it’s a sad mess to see. This happened to one of my hives in the past and the best I could do for the colony was to merge them. At some point all their effort was devoted to cleaning up and corralling larvae and moths - getting stuck in webs in the process, and starving since they were unable to forage and stores had dwindled to nearly none. I don’t know if foragers stay in to help with moth duty, or if bees that would become foragers never do because they have to keep battling the infestation 