I bought a cheap CP recently to play around with, and noticed an interesting phenomenon:
Using it with my 24mm lens on full-frame, images turn out tack-sharp - there is hardly any difference in sharpness/clarity with and without the filter.
However zooming in to longer focal lengths, I notice that the image quality degrades significantly. At 50mm you can begin to notice that images are not quite as sharp. At ~100mm images are significantly blurrier, and out to ~200mm the filtered images are pretty much unusable.
What about a circular polarizer would cause varying optical performance dependent on focal length? Is this something that higher-end filters can solve, or is it a physical limit (like diffraction limiting?) at play?
Hard to know from your post, but note that a CP filter will loose you between 1 and 2 stops of light, depending on it's setting. This significantly reduces the amount of light reaching both the sensor and the all important autofocus system.
If you have a variable aperture lens, this means that longer focal lengths have a smaller maximum aperture and already let in less light than wider settings. This compounds the loss of light from using the CP.
If there is not enough light for the autofocus system to work well you will find significant loss of focus accuracy.
In general you should remove any filter, especially a CP, when shooting unless you specifically require the effect the filter provides for a given shot.
The problem is the cheap CPL filter, not CPL filters in general.
I have also noticed significant image degradation at longer focal lengths, but only with low quality CPL's. When I use good quality CPL there is no image degradation.
A polarizing filter is likely the most useful accessory you can own. It works by limiting light rays that transverse the lens to just one direction of vibration. We use them to mitigate reflections. Additionally the polarizer acts like a UV filter in that it cuts haze seen in distant landscapes. The polarizer enhances clouds and intensifies sunlit vistas without altering the color of the subjects.
The bad news is a polarizer can (not always) interfere with some camera automation like auto focus and exposure determination. This is true because many cameras incorporate polarizing filters in their systems. An external polarizer might interfere.
The polarizing effect we need can be mitigated as to its interference with camera automation if we sandwich a standard “linear” polarizer with a second filter called a “retarder”. This combination is considered best as the retarder un-polarizes the light. In other words the first polarizer does the trick; the second retarder alleviates the harm to the automation.
As to poor image quality, one filter is degrading, two filters stacked together is worse. It takes good quality workmanship to make a circular polarizer.
It's highly likely that cheap CPs use materials that are less optically clear. The camera uses a smaller area of the lens/filter when zoomed in, therefore magnifying the optical artifacts of the CP, which results in less detail.
This, combined with the fact that each lens has a sharpness "sweet spot" at a specific f-stop, and detail is lost to greater degrees the further you move the f-stop from that sweet spot (in either direction -- up or down). The CP causes you to open up the f-stop one or two stops, which may move you further away from the sweet spot.
I need to come back later with a diagram to help with this.
It's not just polarizers. It's any crummy filter. Polarizers have more surfaces to get right, so it's easier for this problem to show up.
Let's suppose that your filter isn't perfectly flat, but has a wave in it. That wave compared to the flat on the other side makes it a lens. A very very mild positive diopter close-up lens. The the light that went to that part of the filter is going to be in focus in front of the sensor.
To keep the arithmetic simple, compare a 20 mm lens and a 200 mm lens.
Your wide angle, remember has huge depth of field. So a filter that nudges different light paths in different directions doesn't matter. Also that wide angle is compressing a lot more world onto a same sensor, so distortions added by the front filter get shrunk.
This is a simplified explanation, but it gets the main idea across.
TL;DR: Test your polarizer on your longest lens of that filter size.