That sure looks like the word subsidy to me? Now back the claim up.
Using the word 'subsidize' in a sentence does not equal saying 'New York State subsidizes gun permit administration costs'.
What that paragraph said was, in effect, 'NYS charging $10 for a gun permit doesn't mean it costs NYS $10 to process. States subsidize administrative fees in some cases, and that may be what's happening here'. Can you see how that is not a declarative statement like 'the gun fee is unreasonable' is? I can't believe this needs to be explained.
I've got places to go now, so I'll be unable to continue this lovely conversation with you. Please do consider reading the opinion though, you might find it enlightening.