"Check that mate" : OED on usage of 'checkmate'

In a recent post How exactly is “to checkmate” used as a verb?, the answer given by @WS2 listed some OED examples of metaphorical usage of 'to checkmate'. Among them:

[1649 A. Ascham Bounds Publique Obed. 58 At this distance he [sc. Jas. I] contrived how to extinguish or check that mate [sc. the Kirk] there.]

(emphasis mine; cannot access OED; unlike other given examples enclosed in square brackets)

I find this example peculiar. In what sense does 'checkmate' occur as a verb here? And what is actually going on in this sentence? Is the word mate as occurring here cognate with -mate in 'checkmate'?

Can anyone shed some light on this? (I cannot even decipher the source.)

This citation is listed under an entry for checkmate in OED which reads:

2. transf. To arrest or defeat utterly, discomfit. In mod. use, often: to defeat or frustrate the ‘game’ or scheme of (any one) by a counter-movement.

This at least tells us which meaning of checkmate this is supposed to be relevant to (but not a direct example of, based on the OED's use of brackets).

I cannot explain the use of mate but it would appear that the Kirk in the note immediately after is the Church rather than a person.

If I were to speculate I would guess at a play of words between checkmate and this meaning of mate, from OED or some variation thereof:

†2. A suitable associate or equal adversary; an equal in status. Obs.