The whole reason push for Black power is to push forward to black nationalism. They are making an effort to make people back their ideas more, and want people to feel more patriotic to their ideas. They are not getting a whole lot of success in pushing this movement though.
Black Power does push for black nationalism because its aim is to put as much power as possible in the hands of black Americans. Leaders such as Jesse Jackson and Al Sharpton are aggressive, but are not even as ardent as true black power elites, such as those in the Black Panther party that push for blacks to rule over all ethnicities.
Black supremacy does not seem to be very strongly linked to black nationalism, largely because there is such as well known racist trope involving sending blacks back to Africa. The US is home to everyone - not just whites or people from Europe, and there is no need for further sub nationalisms.
No, I do not believe that Black Power is about pushing for black nationalism. I do, however, believe that it is all about trying to make black voices heard and taken as seriously as the white blob majority's voices are heard and heeded. We still have a long way to go for race relations.
If anything, Black Power pushes for more blacks to be leaders to black people in the United States. There is no such thing as "black nationalism" in America because the United States comprises many ethnic groups. Black Power is more about giving blacks the freedom to be who they want without constraints.