What's your guess as to what it was struck through? From the shape and size and the fact that the design still shows through, I would guess cloth or paper. Not thick enough to obliterate the design but thick enough to show that something was there. Truth is, we'll never know.
It's my guess that grease & debris compacted in some area of the bust by continuous striking fell out of the die and onto the planchet. I've seen this happen on plenty of Lincoln cents, but I've never seen one (strikethrough) this size on a Kennedy. Chris
Send it to NCIS. They will tell you what it was struck through and a history of when, who, where the object was made and anything else you might like to know.
OP, it could be struck through paper or cloth but it could also be grease and/or debris. At ant rate it's very nice and a great find. Thanks for sharing.