If you are writing an essay where you have to prove a point, you would give that point. When you elaborate on that point, that is called support. You are giving more reasons why you are right.
I would say that "support to" is used to mean a literal support such as to hold up a roof. He added a steel brace to give more support to the roof. I would say "support for" to indicate approval for...
She wants you to be there for her to help her through whatever she's going through. She wants you to be there for her to be someone to give her that extra push and to be someone she can go to, to...
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