Hi Maureen,
It really doesn't matter how you give the tidbits, the important part is the time between the click and the tidbit. Generally, for a reward to be associated with a particular behavior (with dogs, at least), the reward has to come within about 3 seconds of the behavior. The clicker gives you more time to reward by "marking" the correct behavior and giving your time to get the reward to the bird. High-level tidbitting, tossing to the ground, whatever you prefer.
I kind of like tossing the tidbit, because it gives you a chance to "reload" with a new piece while the hawk is busy getting the tidbit. You can click and toss--the reward is presented almost immediately, and the bird still makes that association with whatever behavior you just "clicked."
Just remember that when you click, have a reward ready. Click = reward to the bird, so if you click and DON'T have a reward ready, you can confuse the bird in regards to the meaning of the click. If you need to space out the behavior you are trying to capture/mark, it's another good reason to toss the tidbit
If the bird is on the fist, I usually palm the tidbit in the same hand as the clicker. If the bird is on a perch, I hold the clicker with one hand and a tidbit in my closed fist with the other. With my imprint gos, I have a dish of tidbits in plain sight and just grab one out of there to toss or hand-feed. I don't know that I would do that with a passage redtail though...
~~~Ally~~~ Missoula, MT
If you dislike a person, walk a mile in their shoes. Then, you are a mile away from them, and have their shoes.