Let me start from the beginning of the question which mentions a "scanned copy of my written signature". I used a picture taken with my phone. By the way, I recommend using ink or a strong black pen to make sure the written portion has clear lines that stand out from the white paper background.
Step 1 - Make a transparent signature image
You'll only need to do this once. For this I used gimp, which you can install on Ubuntu with:
sudo apt install gimp
You might want to start by getting a nice rectangle with only the signature. For this I used the Rectangle Select Tool, then Edit > Cut, and Edit > Paste as New Image.
Next I followed the steps to make the background transparent by using the Fuzzy Select Tool and pressing Delete. This gets rid of the white background (important because in my case the picture didn't come out with a clear white, was more like grey) leaving only the black signature. I then exported this to a PNG file. Now I have a signature file that I can reuse!
Step 2 - Insert Signature in PDF
Once you have a transparent signature it makes it easier to use a variety of tools. I used GIMP again. I was given a Word document so I used LibreWriter to convert it to a PDF first. Then I opened it with GIMP and accepted the default of each page as a layer.
Next I opened the signatured file using File > Open as Layers. I used
CTRL+S to resize it and then the Move Tool to position it nicely on the page. Then I right-clicked on the signature layer on the right and selected "Merge Down".
The final step was to use "File > Export as" and giving the file name a PDF extension. In the options dialog I accepted the default of "use layers as pages" and also had to tick "reverse page order" to get the order page right.
Note: I had done this before on Windows and it was a lot easier with Acrobat Reader, because I already had the transparent image. Although the Linux method is a bit more involved, it's not that much more complicated if you follow the steps, once you have a transparent PNG. Any suggestion for improving this answer is welcome.