Since this is presented as a direct quote, I presume that Time's editors did not try to correct grammar. In other words, the ambiguous meaning resulting from the placement of "very badly" is likely all Trump's doing. Notice, though, that there's no ambiguity if we put it in the "split infinitive" position: "I want very badly to debate."
True, unreconstructed pedants will note you can also say "I very badly want to debate." But that's not the point. The default position (having "very badly" at the end) creates an ambiguity. The so-called "spit infinitive" avoids the ambiguity without the formal tone of placing it in front of "want." That sort of formality would not be in keeping with Trump's ordinary way of speaking. Yet more evidence that the split-infinitive prohibition has absolutely nothing to recommend it.