“Through shaking voice and hands, I barely managed to give a speech.”

If you can change the words a bit, I’d make it:

“With shaking hands and voice, I barely managed to make it through the speech.”

Firstly, a speech is ‘made’, not given. Secondly it was not ‘through’ but ‘due to’, or ‘ because of’, whichever you prefer.

Thirdly we talk about a ‘quaking’ voice and trembling hands.

Finally I would suggest you use ‘my speech’ instead of ‘a speech’ .

Try, “I barely managed to make my speech because my quaking voice and trembling hands.”

I hope I have not offended you by pulling your sentence to pieces!!

You could also change through to: for my