Americanism "to care SOME about something"

Solution 1:

As a native Californian I have no trouble understanding that sentence but I don't think I'd ever actually use it in conversation. I think it's most likely an awkward construction that comes out of social science jargon. Poll questions commonly take this kind of form: "How much do you care about Climate Change? Circle one: 1) Not At All, 2) Some, 3) A Lot". Sometimes this is modified into awkward sentences, like so: "How much do you care about Climate Change? Circle one: 1) I care not at all about climate change, 2) I care some about climate change, 3) I care a lot about climate change." Pollsters care more about perfectly standardized parallelism than natural phrasing because the point is to assign potential answers a quantified score that is then amenable to statistical analysis. Also, when something is part of the jargon of your profession, it tends to sound completely natural to you even when it is not standard colloquial English.

Speaking of jargon: This kind of question uses a "Likert-Type Scale," which is a variation of the "Likert Scale," an extremely popular survey question format that is widely used in everything from political polls to personality tests. The classic Likert scale asks you to rate how much you agree with a statement from 1 to 7, like so:

I care about Climate Change:

1 Strongly Disagree
2 Disagree
3 Somewhat Disagree
4 Neither Agree Nor Disagree
5 Somewhat Agree
6 Agree
7 Strongly Agree

Examples of Likert-Type scales from Clemson University

Edit: In this graph from the report cited by the OP, we can clearly see the standardized, fill-in-the-blank nature of this poll question:

Solution 2:

You should note that some there is short for to some degree or to some extent. It doesn't necessarily mean some is used as an adverb.

There are following choices; A great deal, Not too much, Some, Not at all.

Some just indicates a degree (extent). It doesn't necessarily mean it is used as an adverb in a given sentence.