In the above video 2016 State of the Union Address, at 2:10, according to the script, president Obama said "I’m going to try to make it a little shorter. ", but I cannnot hear " I'm going", is there a special pronunciation rule?
There is no pronunciation rule, for this
What you are hearing is
Ummm gonna try to make it a little shorter
where Obama is being a bit more informal in pronunciation with gonna and Ummm and a slight folksy accent. His cadence was also faster. He's doing this since he trying to get a laugh as it is his last SOTU. Notice the difference in his tone of voice when he's joking about "getting back to Iowa" and then when he resumes his set speech.
This is in contrast to his opening remarks which are clearly enunciated and measured and what everyone would normally expect.
At the time posted, President Obama appears to say:
Which is basically just a quick way of saying
The best way to think about it is like a contraction:
I'm+gonna = I'mna
Which is just slang for:
I'm going to
It is not a word of course, but it is basically just him saying "I'm gonna" very fast and in the process cutting out half of gonna. As someone mentioned in the comments it sounds very natural to a native speaker because that sort of "sloppy talk" is very common in informal conversations.