And so we conclude that for a character with only Sponsored Magic, no Evocation or Thaumaturgy, one probably can't take Refinement specialisations, only item slots.
Very probably, but not explicitly stated, no way to define further short of definite Word of God, which doesn't happen, I believe.
I gather that a related question is whether your sponsor's magic theme qualifies as an 'element' in evocation, or a theme/field of application in thaumaturgy. It then leads into whether you can take specialisations in your form of sponsored magic if you already have full Evocation and/or Thaumaturgy.
In addition, if you already practice evocation, you may use a sponsored power source to “supercharge” an element you’ve already specialized in. So Summer magic might combine with the air element to give a “breath of life” effect; hellfire might combine with fire to produce, well, hell-fire; and Kemmlerian necromancy might combine with the spirit element to inflict potent visions of death upon a victim. This sort of combination allows the spellcaster to use his existing evocation specialization bonuses with the new power source.
I believe the only useful part of this is the last sentence of the paragraph, where what they mean is that if you have +4 to offensive fire or spirit evocation, you may apply the +4 bonus to when using soulfire to attack.
Which suggests that your Sponsored Magic theme acts like a free supplementary element that does nothing except colour your magic - the soulfire-boosted attack would do the Catch-satisfying, Toughness-downgrading thing it does, plus your standard fire attack shifts, and that's it.
And it seems to imply you still can't take Refinement specialisations for Sponsored Magic, because it wouldn't make sense for you to be able to do that and still have the book give this roundabout way of getting your bonuses. It does mean you can just pile your Refinements into the relevant elements for using your sponsored theme with, though, getting the same general effect.
Most of the Sponsored Magic themes we've seen can easily fit into one or two of the five elements Harry and co. work by, so this shouldn't be a major hurdle. With hellfire and soulfire, it's even been explicitly noted that despite the -fire description, as the primal forces of creation and destruction, you can apply them to any element. There isn't even a problem, with those two.