Do you have an explanation for those specific cases? Because if so, it would be possible to apply it to every prayer.

The questioner was not asking from the standpoint of reality—whether we see intervention or not—but from the standpoint of possibility: can God intervene or not? From the standpoint of possibility, clearly God can intervene in the chain of causes.

Indeed.

How can such a possibility exist? After all, the whole chain of causes is something that, although not foreseeable in advance, does exist. I can understand how God can intervene in things where there is no determinism, such as human consciousness (in my opinion), but not in nature.

In my opinion you are mixing different planes of discussion. God created the world with its laws. The One who forbade is the One who permitted, and whoever created the laws can also freeze them, nullify them, or change them. So there is no principled obstacle to the Holy One, blessed be He, doing whatever He likes, with the laws or against the laws.
But what then? He chose not to do so. He could have created a different world, but apparently He wanted a world that runs according to laws. How do I know? Because I see around me a world that runs according to fixed laws.
On the other hand, it is impossible to rule out the possibility that the Holy One, blessed be He, intervenes from time to time, because as I explained, determinism is a result of our observation of the world (that we see it operating according to laws), and not of some absolute or a priori consideration. Therefore the boundaries of determinism extend only as far as the observations reach. And as stated, observationally one cannot rule out sporadic interventions that we did not notice or did not see.
And as an aside, I do not see why intervention in human consciousness seems easier to you than intervention in nature. A person's actions too are the result of his choice and his actions. Just as the Holy One, blessed be He, does not intervene in nature, so too He does not intervene in our deeds and decisions, perhaps except for sporadic situations. It makes no difference at all whether you are among those (like me) who believe that human beings do not operate deterministically.

Suppose that theoretically some miracle occurs. When we try to analyze the causes of the event's occurrence (if we are clever enough), we will be able to identify its entire chain of causes.
Such an ability gives us, after the fact, an "ability to predict" the miracle that happened, so prayer has no effect on changing reality.
Is it possible that in physics there is some point at which two events can occur with exactly the same probability, and it is only a matter of luck which one will happen? (There we could point to God.)

As for human consciousness—if you believe that human beings do not operate deterministically, we could say that God intervened a little in a person's will or thoughts. In a pictorial sense, God is the light bulb that turns on for a person in cartoons when an idea "pops" into the character's head.

If in principle the world is deterministic, then there is no room for divine involvement. You do not need an actual ability to predict for that. There is such a point in physics, in quantum theory. But it is on too small a scale (relative to human actions), and there too there is a random distribution given in advance, so there is still no room for divine involvement.
As for the possibility of God intervening in our choices, that is a point Oren raised here on the site. There is indeed such a possibility, but in my opinion it too is strained. You should search the responsa section and find the discussion with Oren.

Thank you very much! You helped a lot!