Is it possible to train a public figure not to use "um" or "you know" when speaking?

Sure.

Toastmaster International.

https://www.toastmasters.org/resources/public-speaking-tips

I took a course in public speaking once as part of supervisor training. The instructor used what I called shock therapy to break a speaker of bad habits. Example: each student had to give a speech in front of the class. One of the things the instructors had told us was body positioning. Your forearms and hands had to be parallel to the floor at all times. One student kept letting his arms drop without realizing it. So the instructor, without warning, took a book and slammed it on the table every time the hands left the desired position. The booming sound was a really quick reminder. He had other similar things for the “um” and “you know” actions. I think with “you know” he would loudly say “I don’t” after every time it was said.

All this led to some pretty comical presentations. And it was effective too. The guy with the hands got the book slammed about 8 times in a 3 minute speech. His next attempt got zero book slams.

If they are already a public speaker they should have already learned.
I learned it in 11th grade.

Of course. It doesn’t have to be a public figure. I could train myself (and I do say “umm” a lot).

I was being trained to be a trainer at work. The guy who trained us gave us an exercise in teaching a skill to new employees. He videotaped our presentations and then we had to watch them.. I was surprised and embarrassed at how often I said “uh”.

It quickly broke me of the habit.

Had a supervisor once that said “um” and “uh” a lot. One morning at the morning brief, I noticed one of my fellow technicians making marks on a piece of paper. After the meeting (15 minutes) I asked her what she was doing. She showed me that she made a mark every time he said “um”, “er”, or “uh”. She had 400 marks. It was astounding, but probably true. He would say sentences like “We, uh, want to, um, do, uh, uh, a better job, er, at this, um task.” Picture that for 15 minutes.

Uh!

Yes, however, I believe some people (politicians for sure) do it intentionally to seem more approachable, genuine, or if they want an answer to seem unrehearsed.

Obama did it to pause to be sure that what was in his mind was what he wanted to come out of his mouth, giving.him a moment to regroup if necessary.