Type about:addons<enter> in the address bar to open your Add-ons Manager.
Hot key; (<Control> [Mac=<Command>] <Shift> A)

In the Add-ons Manager, on the left, select Extensions.
Remove “ Yandex “ Anything.

Also remove SafeSearch, SearchAssist Incognito, FileConverter Addon, SearchLock, DiscreteSearch, WebSearch, Findwide Search Engine, securedsearch

Now do the same in the computer’s Programs Folder.

Windows: Start > Control Panel > Uninstall Programs.
Mac: Open the Applications folder.
Linux: Check your user manual.

Type about:preferences#search<enter> in the address bar.
You can select any search engine to be your default from here.

You can also remove any engine from here.

If what FredMcD posted about (an Extension for Firefox) continue on to my thoughts ....

If Yandex is one of the default / One-Click Search Engines that comes with your localized version of Firefox (many times a different 'default' is set for non-US versions / non-English languages who may use a different "default" by global zones based upon over all popularity in the 'zone'), those "One-click's" are installed afresh with every Firefox update, full version update or ##.dot.# minor (Chem-Spill update like 78.0 to 78.0.1).

But Firefox does respect the user when they disable a search engine here - about:preferences#search - under One-Click Search Engines - click to get rid of the green check-mark to disable a search engine and never see it appear in the URL Bar drop-down and elsewhere in the UI, too.

Unfortunately, Firefox doesn't always remember that user setting after an update, and the user needs to go back in there to disable it again, thus "hiding it". I have never had that issue myself, but have seen threads about that after a major version update occasionally. Typically a one-time thing, but who knows. At worse a minor inconvenience, and easily corrected.

BTW, those One-Click Search Engines do generate per-use income for Mozilla when they are used. income that Mozilla needs and uses to "keep the lights on" and "bills paid" for their operations around the globe. So they protect that against the "One-Clicks" being tampered with by nefarious software that might change them (yea, that happened some before like 2015) and then after Yahoo was sold in 2016, and Mozilla didn't "comes to terms" with the new owner. That was when Mozilla came up with the idea for separating the One-Clicks from the user installed search engines, and made it possible that thru one simple "minor version update" and to "lock down access" to any one but Mozilla. Never affected the user's own choice overall, but nothing works 100% correctly all the time, and still doesn't other than the occasional 'glitch' where the users choice gets reset, for yet to be learned reasons.

Type about:preferences#search<enter> in the address bar.
You can select any search engine to be your default from here.

Thanks for the help. The only thing that worked is completely removing yandex from search engine list. Making Google default search engine (I set it default a long time ago), unchecking yandex, moving it to the bottom and Google to the top, all of this didn't help. Funny thing, though: no "Search in Google" link suddenly appeared, despite it set default and enabled in the list. So, as I see it, still some nasty advertisement.

Hi Крафт, do you see the tiny gray "push pin" next to the icon for that entry in your original screenshot?

Based on that, I think it was the Top Sites list. By default, when you click the address bar in Firefox 75+, before you start typing anything new, Firefox shows a list of your first 8 Top Sites from the Firefox Home / new tab page. Perhaps you had used a workaround to avoid seeing this list in Firefox 75-77, but those older workarounds don't work any more.

If you don't want the list to open automatically:

You can uncheck the box for Top Sites on the Options/Preferences page under Privacy & Security, Address Bar:

Assuming you want to keep the list and take control of it:

In order to remove items from that Top Sites list, you need to make your edits on the Firefox Home / new tab page.

If you use Firefox Home as your home or new tab page, you an open it by clicking the "+" to open a new tab, or by clicking the Home icon. If you use a different page, you can type or paste one of these addresses in the address bar and press Enter to load it:

• about:newtab
• about:home

Usually the page has a search box followed by the Top Sites section. If your Top Sites section is missing, use the checkbox on the Options/Preferences page to turn that back on: Customize your New Tab page to hide or display content.

When you hover your mouse over a "tile" in the Top Sites section, a little button should appear. Clicking that button will provide options to unpin or dismiss that site from your Top Sites. Unpin lets the site float into natural order in the list, based on your history. Dismiss blocks that URL from ever appearing in the list. More info in this article: Customize your Firefox New Tab page.

Here's an example from the U.S. English version of Firefox:

I suggest taking away all the irrelevant stuff here to free up slots for your actual history, or you can pin, rearrange, and add sites you want in your top 8.

If what FredMcD posted about (an Extension for Firefox) continue on to my thoughts .... If Yandex is one of the default / One-Click Search Engines that comes with your localized version of Firefox (many times a different 'default' is set for non-US versions / non-English languages who may use a different "default" by global zones based upon over all popularity in the 'zone'), those "One-click's" are installed afresh with every Firefox update, full version update or ##.dot.# minor (Chem-Spill update like 78.0 to 78.0.1). But Firefox does respect the user when they disable a search engine here - about:preferences#search - under One-Click Search Engines - click to get rid of the green check-mark to disable a search engine and never see it appear in the URL Bar drop-down and elsewhere in the UI, too. Unfortunately, Firefox doesn't always remember that user setting after an update, and the user needs to go back in there to disable it again, thus "hiding it". I have never had that issue myself, but have seen threads about that after a major version update occasionally. Typically a one-time thing, but who knows. At worse a minor inconvenience, and easily corrected. BTW, those One-Click Search Engines do generate per-use income for Mozilla when they are used. income that Mozilla needs and uses to "keep the lights on" and "bills paid" for their operations around the globe. So they protect that against the "One-Clicks" being tampered with by nefarious software that might change them (yea, that happened some before like 2015) and then after Yahoo was sold in 2016, and Mozilla didn't "comes to terms" with the new owner. That was when Mozilla came up with the idea for separating the One-Clicks from the user installed search engines, and made it possible that thru one simple "minor version update" and to "lock down access" to any one but Mozilla. Never affected the user's own choice overall, but nothing works 100% correctly all the time, and still doesn't other than the occasional 'glitch' where the users choice gets reset, for yet to be learned reasons.

Yes, Yandex is not only "default", but also very intrusive in our localized version, to the form of big "DOWNLOAD FIREFOX WITH YANDEX" button on the top of firefox.com website, with a button "download firefox without yandex" hidden lower on the page. And I still see such things as a very intrusive and bad practice advertisement: as I said above, there was no other way to disable this line, but to completely remove yandex from the search providers list: unchecking it, making google default etc. etc. — nothing works. And when I removed it, no "Search in Google" or "Search in Duckduckgo" line magically appeared on the same place. So as I see it, this line was made specially for Yandex, it was made to appear at the very top right after the user clicks search/address bar. And it was made without letting users know how to disable it if they don't want to use Yandex search. I understand, that Moziila needs to make money, I understand, that nothing comes for free. But such advertisements are just a really bad practice.

Hi Крафт, do you see the tiny gray "push pin" next to the icon for that entry in your original screenshot?

Thank you! I don't use Firefox home page, I prefer blank pages and bookmarks sidebar, and I had no idea that "Top sites" feature (which I found really useful, except for this search line) could do anything with Firefox home page.