to contribute ... to provding ...

to PROVIDE. "money" is okay

I'm not sure you really need "their spare" in there, since people generally contribute only what they can spare. You can, however, leave it in, it is not incorrect.

I would add a comma after "poor". Since "expenses" is plural, you should change the "is" that follows it to "are"---instead of "living expenses per day is", change it to, "living expenses per day are".

You can either change "providing" to "provide", as in this example:

...asking Americans to contribute [their spare] money to provide loans to China's poor, people whose per capita living expenses per day are less than USD 1.25.

or

you can change "to" to "toward" and leave "providing", as in this example:

...asking Americans to contribute [their spare] money toward providing loans to China's poor, people whose per capita living expenses per day are less than USD 1.25.

I think that either of those examples would be correct.