Windows XP, Professional version, Service Pack 3, 32 bit. Audacity 2.1.2 obtained from .exe installer. Llame v 3.99.3 for Windows and Ffmpeg-win 2.2.2. Adobe Flash Player 21. Firefox 45.0.1 with the following extensions, YouTube Flash Player 1.6.2, YouTube HD Plus 1.5.0 and Youtube No Buffer Google is the search engine.
Older versions of Aucacity, Llame and Ffmpeg were first uninstalled before installing the current versions. I believe all the above are up to date versions.
The computer plays music from CDs and from folders in My Music through Windows Media Player without any problems

The problems occur when I play music from videos in YouTube. First, when using YouTube’s default HDML5 player through Firefox 45 which also defaults to HDML5 player, there are so many sound interruptions that it can’t be used. Second, installing Adobe Flash Player and the YouTube Flash Player and YouTube No Buffer, allowed the video to be played with only a few sound interruptions, mostly at the start and occasionally no sound interruptions at all. The YouTube HD Plus extension improves the quality of the sound.

Third, when starting Audacity and clicking on the record button, the cursor moves across the screen to the end of the screen and bounces back about 3 seconds and constantly repeats the process. With the first video, the sound waveform follows the pattern of bouncing back from the screen edge about 3 seconds throughout the recording and usually an excellent recording is made. Then, there is at least a 90 second delay without any sound before the second video can be started and it’s sound pattern starts to appear on the screen. By then, the cursor has moved back to about 3 seconds from the screen edge and is only moving about one second before bouncing back ie. it does not move over to screen edge. There are a lot more sound interruptions in the second song and sometimes the cursor will freeze for a short time then start again with the same one second bounce back pattern and occasionally the cursor will freeze near the end of the song and that will be the end of the recording session. If I should continue with the third video, the cursor will continue this one second bounce back 3 seconds from the screen end and will certainly freeze about one minute into the third video.

I should also note that the drivers are up to date and all equipment is working and there is no malware or viruses in the computer.
There are sound interruptions whenever the mouse pointer is moved when playing YouTube videos with or without Audacity being used.
Any suggestions to resolve or partially resolve this issue would be appreciated.

You are obviously used to installing software. Assuming you have copyright holders’ permission to record the YouTube videos, why not install one of the many YouTube extensions that let you downoad the video? If you do want to edit the audio in Audacity, install FFmpeg then drag the video into Audacity.

If you prefer not to see the current recording point, Edit > Preferences…, Tracks section and uncheck “Update display while playing”.

You don’t say what recording source you are choosing in Audacity. Is it stereo mix? It is not Audacity’s fault if the stream is interrupted by problems on your computer. If you update to Windows 10 your computer will be more secure and the operating system more lightweight and you can use Windows WASAPI loopback recording to record music from YouTube. I would read it that your computer is under-resourced if it cannot play HTML video. One tip that may help is to reduce the number of browser tabs you have open.

If your computer does not support Windows 10 you may install a lightweight Linux distribution like Lubuntu where you can record computer playback with what is called pulse monitor.