Both power as ability and "may" power are described in the everyday speech using the word can, which creates some confusion. In order to clear it, we need to take into consideration circumstances of the specific situation. For example, "I can speak English" is a form of power as ability when we refer to what languages we understand and communicate with. For example, I personally can speak Russian, Spanish, French, English, and German. In contrast, when Indigenous children were taken from their parents and put in boarding schools, they could speak English in the classroom, but they could not speak their native language. In that case an Indigenous child could say, "I can speak English" referring to the fact that she is allowed to speak this language as opposed to her family language.
Being allowed is not the same as being able to do something. If we are not able to do something (no power as ability), we just cannot do it. If I cannot lift a really heavy stone, I will simply not take it off the ground, no matter how hard I try. In contrast, if I am not allowed to lift a particular stone (e.g., because it is sacred), I might still be able to do it if I choose to, but then I will have to face consequences.
I describe "may" power as being allowed, which sounds like there must be somebody to allow or disallow an action. But "allowing" does not have to be done by a person. Laws of nature also limit what we can or cannot do. As the saying going, one can do many things, but some of them only once. For example, I am able to jump off a cliff (I can physically do it if I choose to), but if I do it without any appropriate equipment (e.g., bungee-jumping), I will hit the ground and die. So when I say, "I cannot jump of a cliff without any appropriate equipment", I refer to the lack of "pay" power rather than the lack of power as ability.
In the case above, the limits of my "may" power are determined by laws of nature. But, of course, we are often allowed or not allowed to do something by other people. When somebody does not allow us to do something, they exercise their power as influence (doing), which determines my "may"power, or the lack thereof. For example, I can make a decision to walk down a busy city street naked, but I am really not allowed to do that, so I will get arrested an face the consequences (in this sense, Lisa Stansfield's music video from the 90's does not seem particularly realistic).
As with any other kind of power, "may" power always coexists with some degree of powerlessness. Nobody has an absolute "may" power in a sense that nobody is allowed to do absolutely anything (even the most absolute monarchs in the human history).
On the other hand, we can often observe how people have different amounts of "may" power, which can be described as a form of inequality. Long time ago, I read a Chinese ancient saying which sounded something like this: a poor person steals and goes to jail, a rich person steals and gets richer. Another example that come to mind is that, according to some reports, people with different skin color and cultural backgrounds get different sentences for the same kind of crime (e.g., using illegal drugs). A historical example comes from the book about Louis XIV that I read to better understand absolute monarchy. Louis XIV authority and decisions were challenged many times by different people. Noblemen could be punished for such a rebellion, but not the same way as people without the noble status. For example, in Chapter 4 we read: "While humble Ormistes were broken on the wheel, the Prince de Conti and the Duchesse de Longueville were allowed, by virtue of their rank, to withdraw to their estates."
We can also imagine many situations when one person is allowed to do something while another person is not allowed to do this same thing. This is also related to choices of other people who exercise their power as influence. My husband is a French citizen, so he is allowed to travel to France without a special visa. I am not a French citizen, and even though I can try to get on the territory of France illegally, I might have serious consequences, because I am not allowed to do that without a visa. In this case, my husband is allowed to something that I am not allowed, and his "may" power is determined by complicated bureaucratic conventions and laws.
Distinguishing "may" power helps us to consider some important questions: 1) Who has "may" power in a certain situation? 2) Why does one person have "may" power in a certain situation while another person lacks it? 3) Who is doing the allowing? If it's people (as opposed to nature), what factors are their decisions based upon? 4) How do we know when we have "may" power?
In terms of the conversation about power and responsibility, these questions allow us to avoid unnecessary blame. Somebody might not do something we want them to do because they lack appropriate "may" power. When I was waiting for a decision on my British visa in 2022, I discovered that people in the visa and immigration call power did not have "may" power to contact the decision center in order to understand what is going on or speed up the process.
Another things that these questions allow us to do is to be careful with our actions. Since with "may" power we might find out about not having it only about doing an action, this is essential. If I arrive in a foreign country, I need to be aware of its laws in order not to get in trouble. I cannot use my ignorance about the laws as an excuse if I break one of them. "May" power becomes more complicated when we deal with long-term consequences. For example, in terms of environmental change, many people need to keep doing certain actions for a while (e.g., building plants that pollute the air, creating and buying one-time-use plastic items) before they discover that these actions create big-scale adverse consequences.
Last thing to say about "may" power is that using it feels rewarding, as with any kind of individual power. People enjoy knowing that they can do certain things, and they do not like being told that they are not allowed to do them. This is why it might be inefficient to tell somebody who is littering not to do that, because they will perceive it as an attack on their power. Many people (both children and grownups) choose to push boundaries, even break laws, in order to experience their "may" power. It is rewarding to "do what you want."