If say…a mouse jumped on a bald eagle’s back in mid flight, could it support it? What I’m trying to figure out is can large birds support any weight other than their own?
They carry huge fish in their talons. A small mouse shouldn’t be a problem.
Of course they can. Eagles in particular like to nest high, and they must carry caught prey up to their young to feed them. Besides, even if the eagle at the mouse on the ground, the weight doesn’t just disappear because it’s inside the bird. If eagles couldn’t fly after eating, even a large meal, there wouldn’t be any eagles left.
Such as, say, a coconut? 
Or a hobbit? If he promised not to pinch?
Great points, but what if the animal landed on it’s back, in between its wings? Such as, the extra weight is in the center of it’s body, think it would be difficult at all?
I am unaware that coconuts grow in bald eagle native habitat. Of course, carrying a box of Mounds candy bars is not out of the question.
And I guess there really isn’t that much difference if the extra weight is being carried or in between it’s wings, I’m just askin. Also, what about tiny birds? Would they have any trouble supporting extra weight?
If a bald eagle landed on the back of a sparrow, don’t you think the sparrow might have just a wee bit o’ difficulty?
African or European sparrow?
African or European bald eagle? 
Suppose two eagle carried the mouse together
So what you are asking is if birds fly at the absolute maximum limit of their abilities to do so. In a word, no.
That would probably be the best place for any extra weight to be.
Just think, if a bird couldn’t carry any extra weight then it must be right at the limit of being able to fly at all, which means it wouldn’t have any extra power to climb or cope with turbulence etc.
Birds of prey routinely pick up and carry fish, rodents, and other tasty edibles.
A smaller bird such as a sparrow wouldn’t be able to carry a rabbit of course, but some extra weight shouldn’t be a problem.
Even extremely tiny birds can carry some extra weight. I worked on a project this summer that involved banding birds caught in nets, including hummingbirds.
As you might expect the bands put on the hummingbirds legs are tiny but then the birds don’t weigh much themselves. They seem to carry them with no visible difficulty.