Joe, from what the OP has said, I gather the light won't work without the zipcord.
Justin, no, you can't just cap the zipcord. You need to get rid of it, and replace it with the proper cable, if you want it to be safe. I'd check that light above the sink and maybe the switchbox that controls that light. Might be in some other outlet nearby too. Hopefully its not a 'flying joint' above the ceiling.
Just found the answer. There is an old under cabinet light mounting literature and now I see a hole where they fed the cord through. Looks like they were having an under counter light turn on at the same time as the overhead.
Thank you all so much for helping me figure this out!
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