Is “cut against [something]” an idiom or just ‘cut’ followed by a prepositional phrase?

I would vote for "idiom", because it's in fairly common usage, and it means something other than what the words themselves mean.

To replace with a single word:

"This is quickly becoming a pattern for the Republican governor that contradicts his core argument to voters.

Like you, I could not find a listing of cut against in a dictionary. However, many dictionaries do list the phrasal verbs go against and cut across (meaning “oppose” and “affect multiple groups” respectively).

I wondered if cut against is either a mix-up of these two phrases, or else a sharper wording of go against, as the word cut can mean “to move sharply toward”, which would be a rough synonym of go in that context.

Indeed, cut against is found throughout literature, but many of those usages are literal and not figurative (as in, “When slicing a roast, always cut against the grain”). I searched for some instances where the phrase was being used in a context similar to your Time quote, and managed to find a few:

I agree that “absolute rights” are often defeated by “compelling interests,” this fact does not seem to me to cut against my claim that our culture is rights-based. (R. S. Markovits, 1998)

Even the two-year lead time .. will likely be insufficient to attract new entrants into a distressed industry with high start-up costs and industry-specific investment. This, too, may cut against allowing the merger of firms in a distressed industry. (First, Fox, and Pitofsky, 1991)

These outcomes cut against the rational-game-theoretic approach that sees regimes as the resultants of nations rationally pursuing mutual gains in variable-sum games. (G. M. Gallarotti, 1995)

These usages seem to indicate the phrase can be used to mean oppose or contradict, so it seems go against and cut against can be used synonymously in some contexts.