I have just taken mine apart as if to replace the bulb.
1. The black plastic piece in the pic, and the spring clip should not need to be removed. I'm not even sure how you did that.
2: Yes, you appear to be missing the central part of the reflector.
The process should be/have been
- if it's a multi colour lamp, set it to white = no filter over the bulb
- remove rubber ring and "glass"
- remove the large part of the reflector; that should be loose
- remove the spring-attached central part of the reflector
- exchange the bulb (which should be loose)
- replace central part of the reflector
- replace the large part of the reflector
- replace the rubber ring and glass.
Unless this is a single colour model (which it appears to be) and unless they had a one-piece reflector. On that I'm not sure. I have never seen one in the flesh, and I don't see any online images that don't have the same two-piece as mine. But I'm guessing that the process would be the same but removing the entire reflector in one go.
So I think the first thing you need to do is replace the spring and the black plastic ring into your lamp, and leave them there.
Here is a pic I found online that shows a part dismantled mechanism. It's not the best illustration of these parts in place but you should be able to see the edge/side of that spring clip.
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