I am currently setting up my Firefox in 24.04 to mimic the 20.04 settings. Long ago I added a search engine shortcut by clicking on the little green plus sign on top of the magnify glass next to the search box. Going to that specific site site no longer presents the green plus sign. However, that shortcut still works in Firefox on 20.04. How can I manually create this shortcut in my new Firefox since I no longer have the green plus sign to click?
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Yes, that puts the search box in the tool bar. But how do you put a custom search engine (URL) as an option in that box? That is an URL that doesn't present the green + symbol.
But I want to add a custom site. I don't want Bing or Google. The site I want to add is "https://www.perplexity.ai/". That site provided the green + long ago but not now. So the search option is still there and works with my older installation. I can't seem to add it in my new installation.
The green + in the search box.
What about this browser plug-in ?
And since they seem to have gone commercial, maybe what you are looking for is no longer offered as a free add-on. 
The support page at Mozilla is showing that + sign, but no indication of how to "fix", other than maybe reinstall the search engine add-on, as @Ironfoot
has already pointed out.
I just did that on UM 22.04 with Firefox 133.0 built-in customization options, and I got what you are looking for:
Some more insights. The plus sign is presented in the Search box for every page tab that I visit except "https://www.perplexity.ai/" !!!
Maybe they have a way to block "search for outside search engine".
If you have the ability to manipulate JSON files, the file containing Firefox's default search engine offerings is this
./.mozilla/firefox/{yourProfileFile}/search.json.mozlz4
You could edit that by removing the ones you don't want and add the one you do want. 
I don't know how to do that, myself.
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Thanks. I know, I had a tough time explaining my challenge. Anyway, I did find an extension " mozlz4-edit", I will give it a try once I can locate the JSON file. It is not in the same location as now under 24.04 Firefox is snaps.
Thanks again. I started with the addon, I was afraid to try editing the file, so I just copied my old "search.json.mozlz4" from 20.04 and all worked perfectly.
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That deserves an "Attaboy!" for both the simplest approach and confirming it works!
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If the website has an OpenSearch tag on the page, it's usually as simple as right clicking the address bar and clicking "Add [name]" (for the default, unified address bar). It was a bit sad when Firefox hid the 
+ symbol.
Fortunately, there's an even easier way. Mycroft collects OpenSearch for easy adding:
I have about 28, and use (many of them) all the time!
Otherwise, the ability to add depends on the HTML having this tag:
<link rel="search" type="application/opensearchdescription+xml" href="https://ubuntu-mate.community/opensearch.xml" title="Ubuntu MATE Community Search">
In theory, if the site is quite niche and doesn't have one, you could craft a simple HTML file + OpenSearch XML file and summon a tiny HTTP server (on localhost) to add it. Here's this site's XML for example. That would be an alternate to editing the search.json.mozlz4 directly.
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