In this case, "That" is introducing an essential modifier; "which" introduces a non-essential modifier. Essential basically means that you have to have the text following "that" in the sentence or you can't understand the basic meaning / the basic meaning changes.
For example:
To get to my house, go to the third one on the left, which is red.
To get to my house, go to the third one on the left that is red.
In the first sentence, you go to the third house on the left. It happens to be red. In the second sentence, you go to the third red house - but this may or may not be the third house overall.
In the given SC here, we already know they're being evicted. Why they're being evicted is non-essential info - we still understand what's happening. So we need which.
Also - you'd remissing a "for" in here. They can't afford the payments FOR the homes. You'll hear people say something like "that they can no longer afford monthly payments for." Can't say this - both because "that" is incorrect and because we can't end a sentence with a preposition (for). If you pull "for" earlier in the sentence, it becomes apparent that "that" doesn't work: homes for that they can no longer afford monthly payments. Needs which.
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Stacey Koprince
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