Could you try this:

(1) Disable the Bing Search extension for now.

(2) Select Bing as your default search engine on Firefox's Options page. See: Change your default search settings in Firefox.

For Cortana or the system search box, there is a different article which I suspect it not relevant at the moment, but just in case: Change your Firefox search settings on Windows 10.

(3) If you want Bing as your home page in every new window, go ahead and open https://www.bing.com/ and then follow the steps in either of these articles:

• How to set the home page
• Startup, home page, tabs, and download settings

You can test the change by opening a new window (Ctrl+n).

Does that fix it?

If you still have a problem when using your desktop shortcut, it might be hacked. The Firefox icon can be set up to launch specific pages at startup (or more specifically, when you use the icon). To check whether that is set:

First, open the shortcut as follows:

• Desktop shortcut: right-click the icon, choose Properties
• Pinned taskbar icon: right-click the icon, right-click Mozilla Firefox, choose Properties

Windows normally will select the Shortcut tab. If not, go ahead and click the Shortcut tab.

You'll see the Target highlighted. On 64-bit Windows, that usually is no more and no less than the following:

"C:\Program Files (x86)\Mozilla Firefox\firefox.exe"

If anything follows after that, try clearing it out.

After OK'ing the Properties dialog, you can test right away to confirm that Firefox now launches only your home page. Either:

• double-click desktop shortcut
• right-click pinned taskbar icon, click Mozilla Firefox

Success?

You didn't mention whether you have already done a malware sweep related to the unwanted search site. Just in case you need suggestions for that:

(1) Open the Windows Control Panel, Uninstall a Program. After the list loads, click the "Installed on" column heading to group the infections, I mean, additions, by date. This can help surface undisclosed bundle items that snuck in with some software you agreed to install. Take out as much trash as possible here. (If in doubt, feel free to post a screenshot.)

(2) You can search for programs that don't advertise themselves so obviously with the scanning/cleaning tools listed in our support article: Troubleshoot Firefox issues caused by malware. These on-demand scanners are free and take considerable time to run. If they finish quickly and especially if they require payment, you may have a serious infection. I suggest the specialized forums listed in the article in that case.