My vote is 'no'. If you don't actually possess any of the powers that grant item slots, you can't gain more through Refinement. Nor can you gain extra elements (if access to an element is part of the deal) or increased skill. In a fundamental sense, you aren't the source of the magic, and nothing you personally do can improve it.
If you have those powers, though, and the Sponsored Magic is an extension of your power rather than its source, then sure.