I have a laptop (Intel Pentium 5405u, Intel UHD 610, screen 1920 × 1080) to which I would like to connect a second monitor. Minecraft (runs at around 40-100fps on Gomme) would then run on the new monitor, and Discord would then be open on the integrated monitor.
Are there any significant performance losses to be expected? I mean, Discord is running in the background anyway, does it have to be displayed in parallel, or do I have significantly less FPS with the new monitor?
That depends on what you want to run on your second monitor. But yes, you will have a few drops in performance. Since you are apparently using the integrated graphics, you will probably be able to live with it. But as long as you only use it as e.g. A Discord monitor, the difference won't be that big.
You then just have less graphics performance for Minecraft that is needed to display Discord. You'd have to try whether that really matters that much, but I don't think so. At least DaVinci Resolve is not really worse if I have something else on the second screen.