You could be right. My theory breaks down on giving Barrack Sr. a way to stay in the country. He went back to Kenya two years after Barrack Jr. was born.
There is one way that your theory would make sense. I am not up to speed on Kenyan history. They were going through upheaval. I don’t know where they were at at the time Ann found herself to be pregnant. Odinga didn’t become VP until later.
It is possible that he considered staying in the US but then saw the changes in his country and decided to go back.
However, remember he already had a son and wife in Kenya.
IF there was a marriage - it was a sham marriage - and he had to be given something to make it worth his while to lie on a marriage application that could cause him to face prison.