Writers love quotes because they add human interest and immediacy, but most people you want to quote don’t talk clearly. You have several options: don’t quote them, paraphrase, use a partial quote, ask the question again or sharpen the answer.
You use a quote because it’s the best way to explain something or to capture character. But quotes require a lot of apparatus (attribution, identifying speakers and context), so you should use them sparingly. Don’t quote just to quote. And apply even more rigor to poorly-phrased quotes. So first, just leave them out.
You can always paraphrase a quote. If you can write it better than the source said it, you probably should. Some paraphrases include short bits of quoted material, what we call a “partial” or “fragmentary” quote. For example, your source says about his mother, “Well, you know, she’s sorta with it, or not, um, in, out of it, um, you know, just occasionally lucid.”
The quote’s a mess, not worth its space or confusion, but you like the way it characterizes the speaker’s frustration with his mother. So you can drop a partial quote into a paraphrase of other things the source said, like this: Jonas’ mother, “just occasionally lucid,” seldom finishes her sentences.
Partial quotes tax the readers’ patience. Readers wonder what the rest of the sentence said, what you’ve left out. Multiple voices in the same sentence always have the potential to confuse. And fragmentary quotes easily become a habit.
Some writers, not including me, will then write what they just said and, if the speaker agrees, punctuate it as a quote. I regard that process as illegitimate, a form of fiction, because it leads to exchanges like this:
If you don’t improve the quote on the spot, you can always call the source later and ask the question again. In my experience, you get a viable quote, and new information.