I guess you have a container installation. During setup you have done something which changed the behaviour of a typical install. I guess you are using the development stream.
The message Not Found (“No such image: ghcr.io/home-assistant/qemux86-64-homeassistant:2023.10.1”), resolves to the following page. The request is http and it should have been https. So something is wrong somewhere there. I suggest a clean install and follow the stable stream.
root@host03:~# ha core check
Error: Can't execute command: 404 Client Error for http+docker://localhost/v1.43/images/ghcr.io/home-assistant/qemux86-64-homeassistant:latest/json: Not Found ("No such image: ghcr.io/home-assistant/qemux86-64-homeassistant:latest")
root@host03:~#
and via web
23-12-17 16:22:52 INFO (MainThread) [supervisor.resolution.check] System checks complete
23-12-17 16:22:52 INFO (MainThread) [supervisor.resolution.evaluate] Starting system evaluation with state running
23-12-17 16:22:52 INFO (MainThread) [supervisor.resolution.evaluate] System evaluation complete
23-12-17 16:22:52 INFO (MainThread) [supervisor.resolution.fixup] Starting system autofix at state running
23-12-17 16:22:52 INFO (MainThread) [supervisor.resolution.fixup] System autofix complete
23-12-17 16:23:16 ERROR (MainThread) [supervisor.docker.interface] Can't install ghcr.io/home-assistant/qemux86-64-homeassistant:2023.12.3: 404 Client Error for http+docker://localhost/v1.43/images/ghcr.io/home-assistant/qemux86-64-homeassistant:2023.12.3/json: Not Found ("No such image: ghcr.io/home-assistant/qemux86-64-homeassistant:2023.12.3")
23-12-17 16:23:16 WARNING (MainThread) [supervisor.homeassistant.core] Updating Home Assistant image failed
23-12-17 16:23:34 INFO (MainThread) [supervisor.host.info] Updating local host information
23-12-17 16:23:34 INFO (MainThread) [supervisor.host.services] Updating service information
23-12-17 16:23:34 INFO (MainThread) [supervisor.host.network] Updating local network information