This isn't a problem with Cycles, the same thing will happen in the standard renderer - that is, if you try to render the current frame to the PV, it won't do so.
This happens because when Cinema renders an image to the PV (but not to the viewport, and not in the RTP of Cycles) it clones the current document then renders the clone at the specified frame. In many cases this works fine but for many dynamic scenes what gets rendered depends very much on the previous frames; data will have been generated and updated in those previous frames that each succeeding frame needs. This frame will then alter the data for the next frame.
If you try to render a single frame to the viewport there's no problem as you're rendering the document present in the editor and all the data has been generated and updated. But if you render that frame to the PV, it will not have that data and so what you will get is essentially whatever was on the starting frame of the animation. Try your scene with the standard or physical renderer, you will see the same result when rendering a single frame.
The solution, if you want one specific frame to be rendered to the PV, is to render the frame sequence from the start to the one you want. Yes, it is a pain to have to do that but it's the result of how Maxon implemented the external renderer (which is what standard and physical are considered to be when rendering to PV and why it's easy to add multiple render engines to Cinema).