I know that it definitely makes sense to add 가 here because 수 is indeed a pseudo-noun. However will there be any differences between adding 가 and not adding? It would seem that they are the same thing, because I heard them equally frequently used.
They are different. But the difference is subtle and I can't find the general rule right now. Let me instead show you some examples. I never studied Korean professionally, so the following is based on my experience.
When we want to express will, we remove 가.
우리는 할 수 있어.
We can do it.
이 경기는 질 수 없어.
I can't lose this game. (I really want to win this game.)
When we want to express complaint or assert something, we put 가.
모기 때문에 잠을 잘 수가 없다.
I can't sleep because of mosquitos.
이 경기는 질 수가 없어.
I can't lose this game. (There is no way that I can lose this game.)
In my opinion, it's better to remove 가 because it sounds more neutral, but they seem to be almost the same in many cases.
I reckon there are some different nuiance between each and I will show you additional cases.
수가 있다. There is/are a/some way (for specific purpose given by the context)
(Korean usually don't concern about articles.)
이길 수 있다. We can win.
이길 수 없다. We cannot win.
I'm afraid to explain them above like this. (Maybe some korean don't agree with me.) But I would say only these sentences give more stress on the verb to win, in contrast, the others do so on the 'way'(수).
이길 수가 있다. There is a way to win.
이길 수가 없다. There is no way to win.
이길 수도 있다. There is a less certain way to win.
비길 수도 없다. There is no way even to draw.
For the meaning of 'even', using to win instead of to draw doesn't make sense semantically.
이길 수는 있다. There is a certain/some way to win.
이길 수는 없다. There is not a way to win.
As shown above, each structure of -수가, -수도 and -수는 is to mean slightly different nuiance.
I think they are the same...
Although I can't explain it, 가 is used more commonly in sentences such as 나는 잘 수가 없어 (I cannot sleep). But sentences such as 나는 이거를 할수 있어 does not use 가 as often. I think it's that verbs that require 를 afterwards do not use 가, as verbs that do not require it use 수.
Sorry, I only know what sounds correct in my head. :(
Source: My mom is from Korea and only speaks in Korean to me.