I've had this issue since the "new" backup system was introduced. Basically any backup (auto or manual) seems to complete (per supervisor logs and backup lists) but HA must not get the message. As the spinner runs forever and it's not possible to make further backups without restarting HA.
I'm running HAOS on a Proxmox VM and have updated Supervisor throught he latest 15.1 hoping it would resolve, but no lock. The one thing that seems supsicous in the Supervisor logs is the following (which others with my same probablem seem to see also).
Note the "No Home Assistant Core respose" items..
2025-04-01 09:57:22.669 ERROR (MainThread) [supervisor.homeassistant.core] No Home Assistant Core response, assuming a fatal startup error
2025-04-01 09:57:22.669 ERROR (MainThread) [supervisor.core] Can't start Home Assistant Core - rebuiling
2025-04-01 09:57:22.671 INFO (SyncWorker_4) [supervisor.docker.manager] Stopping homeassistant application
2025-04-01 09:57:29.257 INFO (SyncWorker_4) [supervisor.docker.manager] Cleaning homeassistant application
2025-04-01 09:57:29.407 INFO (MainThread) [supervisor.homeassistant.module] Update pulse/client.config: /data/tmp/homeassistant_pulse
2025-04-01 09:57:29.548 INFO (MainThread) [supervisor.docker.homeassistant] Starting Home Assistant ghcr.io/home-assistant/qemux86-64-homeassistant with version 2025.3.4
2025-04-01 09:57:29.548 INFO (MainThread) [supervisor.homeassistant.core] Wait until Home Assistant is ready
2025-04-01 09:57:34.892 INFO (MainThread) [supervisor.resolution.evaluate] Starting system evaluation with state startup
2025-04-01 09:57:34.892 INFO (MainThread) [supervisor.resolution.evaluate] System evaluation complete
2025-04-01 10:00:29.814 ERROR (MainThread) [supervisor.homeassistant.core] No Home Assistant Core response, assuming a fatal startup error
If I go into the shell on the Supervisor container I can ping Home asssistant, so it seems ok Docker network wise. Not sure where to head next though?