What happens to automations that run at a specific time of day if HA is offline?

I’m assuming that because of this thread:

The answer is it doesn’t run. This makes sense because in theory, HA could be offline for minutes, hours, days or longer.

Is this still correct ?

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Correct.

As Tom says, it won’t run.

But you could add a HA Start trigger to any relevant automations plus apropriate condition checks then run your actions.

automation:
  trigger:
    - platform: homeassistant
      event: start

I’ve used this in automations to cater for (light duration) Timmers that could Finish whilst HA is down, and other situations. In this case, the timer Finish not firing could leave (lights etc.) on too long.
Sometimes it may be worth triggering a short while after HA Startup to ensure that all is settled down and/or also to avoid a sudden loading.