Merging Two separate Video Clips into One with Final Cut Pro

You can place both clips together in a Compound Clip and they'll act as one single clip. Just select both in the Browser, right-click, Make New Compound Clip.

Put the two clips into a project and export the project.

You will likely benefit by watching these highly recommended free tutorials.

Steve at Ripple Training’s 60 minute FCP tutorial overview: https://www.youtube.com/watch?v=HqUP7Zgeuck

The Final Cut Bro getting started:  https://youtu.be/L3bZ-ETKZ8U

Izzy Video tutorials. Older so the UI looks different but the concepts are still correct: https://youtu.be/_jvyrnQpR1E?list=PLp5SAgbM8S87n1qvv34-PAU2zMHLlXyfb

Without even using FCP, you can merge, trim, or rearrange clips using Quicktime Player. This doesn't work for all codecs, but it normally does: Add a movie or clip in QuickTime Player on Mac - Apple Support

For more details go to Youtube and query on "Easily Trimming Video Files With QuickTime Player"

I realize your question was about FCP, but the above can be a good troubleshooting technique. For some simple tasks, it might be all you need.