Upgrade from Cambridge Audio 840C to lessen glare?

Before you dump your CDP, have you listened to alternative cables. Some Nordost cables can easily take some SS systems over the top. If it ain't that, then I would explore some CDP's with a bit warmer tone. Some of the Marantz units can add a bit of warmth to SS.....Or you can get a tubed CDP.

While reading your post, my immediate thought was, is the amp new? Then I got to the system, and like Newbee, I'm thinking Nordost cables. A few years ago, for lack of a better word, I was married to Nordost speaker cables. I used Blue Heaven and up the line higher than I will admit to with various speakers. Finally, when I tried something different, I realized how smooth extended treble should sound. Some have great results with Nordost, and others, like me, describe them as lean or bright or unrefined treble.

Try isolating and conditioning the ac to your digital. Your new amp is simply be revealing what has always been there.
The better your system becomes the more you need digital isolation. Your player is very good. Just give it the very clean power & isolation needed. You will be very well rewarded.

As you have a somewhat reasonable budget, I suggest stretching a bit and picking up an Ear Acute. It's 6K, but worth every penny, IMO. It has that quality of being smooth, yet it really digs out the details, unlike some other "smooth" players I've tried, like the Lector, that just sounded too dark for me. PLUS - you can tune the EAR by tube rolling. I happen to like the stock tubes right now, but I did try some vintage Amprex tubes and it was a big difference.

I know it's somewhat over your budget, but sometimes you just have to bite the bullet - always within reason of course. You might even get a dealer demo at a decent discount with a full warranty and all. I have the older model, but the newer one has a digital input, which is not an issue for me. Best of luck

It is so hard to helpfully reply to question like this. We don't know what kind of music you are listening to, the characteristics of your room, and haven't heard specific examples of the "glare" you refer to. That said, take all I say with a grain of salt.

Realize that just because common wisdom says that any "x+y" pairing of equipment brands is wonderful, that doesn't mean YOUR ears and brain will agree.

It's possible that several factors mentioned by others are collectively contributing to your impression of glare. It may take addressing some combination of interconnect and speaker wire, power conditioning, digital equipment, and room acoustics to make things gel in a pleasing way. You are reaching the level of equipment quality where it can become very tricky to balance increasing revelation of detail with the desired overall tonal balance. All of your components are reputed to be on the "neutrality and insight" side of the audio spectrum and sometimes it can all just become to much of a good thing. There can be more than one path forward at this point. It could be your issues would be solved with one change: a good tubed preamp. It might just take addressing room acoustics. Very hard to give meaningful advice if we can't hear what you are hearing. It's going to take experimenting with demoing replacements for suspected culprits from someone like Audio Advisor who offer returns or buying used equipment you can hopefully resell with little to no loss if it doesn't work out.