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I've been playing the Allied campaign and I cannot seem to understand when air support is available and when it's not.

Sometimes I'll set a push in motion and then when I go to call in support... it's gone.

Or sometimes I'll just see the air support icons pop up for no reason.

Could someone explain this to me? I'd really appreciate it.
I don't quite understand your second paragraph: are you saying that the air support options vanish without you using them?
As for mechanics, in campaigns aircraft availability is governed solely by the map scripting. You are given air support calls whenever level designer felt it would be appropriate. After that you can use them, you can hoard them. Mind though that aircraft missions are not superpowers as they still can be shot down by the opfor AA, so they merely really close to superpowers. I haven't seen yet any instance of your air support calls being withdrawn by the game without you using them, but I don't own any of the DLCs, so it might be a thing there.