This takes me back. I bought this lens and its big 800mm F8 brother in later 2012 and did a fair bit of testing with them and produced three powerpoint presentations about them. I purchased them because they were cheap, cheap, cheap and I just LOVE bargains! (Just email me on email@example.com and I will email them to you. Total size under 10mb. On both lenses I was worried about precise focus because it is deadly. I had visions of building a little adjustable frame for precise focussing, but finally settled for my Neanderthal option, namely sticking a removable adhesive rubber furniture pad across the focussing ring when I had achieved precise focus. However if the lens cools down significantly, the focus will shift a tad. Some poor fellow took such a lens to a solar eclipse, and, you guesssed it, the focus shifted, and there was a gorgeous longer exposure of the corona, badly out of focus!
I found the 500mm great apart from this focus problem. With the 800 mm, I could not get it precisely sharp at first and sent it back but was told it was within parameters. I feel it was partly the good old focus problem and atmospheric issues, as well as the lens. After all, you cannot expect astronomical quality – you gets what you pays for – if you are lucky!
I dusted off the 500mm lens for the recent July partial eclipse of the Moon because it was very low in the sky and any longer focal length would have been wasted. (image attached). (I used my adhesive rubber furniture pad to keep focus under control.)
Just a thought. If you are concerned about contrast of the image, try to always use the lens hood that comes with it, which you can reverse over the lens (with a bit of padding) when storing it.
I have never had a problem with correct focus across the field as the powerpoint will illustrate, so I can’t help you there. If I had a problem, I wouldn’t trust my neanderthal skills to take it apart and try to fix it, but as my wife will tell you, I am not really a handyman….