Tricky to tell from your photo, but if you’re gong is like this one
https://www.nzhuntingandshooting.co....g-gong-116498/
You could just have a contact that breaks when the gong swings back, and wire the eye light circuit through a NC contact on a basic relay.
You would want the contact behind the front shroud. Or make it more realistic and simple at same time. Wire the eye directly through the sensor for always on, when you score a hit… lights out, at least till the gong swings forward again.
Another option would be to have an optical sensor (like in a shop doorway). When the gong swings back it swings into the sight line of the optical sensor to light the eye.