HA on Raspberry Pi including desktop on Pi

sorry but I`m totally new to Home Assistant.
I got it installed on a Rspberry Pi 4 and all works well.
I can access all from my PC.
Know here ist the question:
Is it possible to run the Home Assistant Desktop also on the Pi?
Would be great to avoid a second machine running?

You probably installed HA OS, and then the answer is no. HA OS runs headless, so does not have a desktop.

If you are willing to loose some features of HA OS, see

You would install an OS with a DE, and use either HA Container (docker), or if you are a python expert, HA Core (venv)