OK, so aside from the obvious suggestion to get your news from other sources....
The reality is what you're almost certainly seeing is temporary and not permanent. The more you watch things that aren't Fox News, the less visible the logo will become.
You could do any or all of the following:
1) Watch other stuff, preferably lots of other stuff... Normally, you can probably watch a single channel with a logo for something like, I dunno, 25% of the time without it getting retained. Since you are at more than 2x that according to your guesstimate, the retention will be a bit more stubborn.
2) Hit the Zoom button when you watch Fox some of the time so the logo moves somewhere else. The logo will split its time this way and be less "retained" in one place.
3) Really focus on not paying attention to it
4) Leave the TV on other content (preferably without a corner logo) when you aren't watching. The more time the TV spends not on that channel, the sooner the logo will go away
5) Try lower brightness / contrast settings, perhaps a somewhat brighter version of the "eco" mode on the TV (which is too dark). If you are at lower than normal settings, the retention will be reduced. Only use that setting when watching Fox News.
Now, that said, if you really watch Fox News that much, then 2/3 of the time, the logo is actually on so it can't really be bothering you that it's "retained". That's a good ratio of not being bothered, I'd say.
On another note, politics aside, Fox News spends a lot of time agitating its audience, getting it angry about stuff "those Democrats are up to." It might be a good thing for your sanity to just watch less of it anyway so that you aren't always so angry about whatever they are angry about today. I'm not say change your politics -- that's really up to you -- but if you watched less Fox News, you'd probably end up better able to calmly discuss your beliefs with people and more open to questioning specific things in them. And really, that's always a good thing. (See my signature in this post and the guy who wrote the quote for more on that.)