Is there a verb for someone trying to avoid the question at hand by digressing. Or to mess up the answer so as to create confusion? Or to give a vague answer?
(At the back of my mind, there is a word that is something like 'subfugate' - which is not an actual word I realised - but something similiar or along those letters? Or I could be plain mistaken)
I thought you were looking for a verb. If you're still looking for a verb:
Prevaricate, evade, dodge.
Consider, fudge.
: (v.i.) to avoid coming to grips with something : to fudge on an issue.
: (v.t.) to avoid coming to grips with (a subject, issue, etc.); evade; dodge Random House
You can also circumvent a question.
circumvent: to avoid or try to avoid fulfilling, answering, or performing The Free Dictionary
What you describe sounds like dodging the question:
Question dodging is the intentional avoidance of answering a question.
(https://en.wikipedia.org/wiki/Question_dodging)
In the context of politics, another word for this kind of trick would be pivot (it's a noun, but I think you can verb it):
Brett O'Donnell is a debate consultant who trains Republican candidates. He has worked with George W. Bush and John McCain, and for a short time earlier this year, he helped prep Mitt Romney.
O'Donnell is an expert on "the pivot."
If you have watched a debate, you have watched a pivot. "The pivot is a way of taking a question that might be on a specific subject, and moving to answer it on your own terms," O'Donnell says.
(http://www.npr.org/2012/10/03/162103368/how-politicians-get-away-with-dodging-the-question)
Pivoting to answer the question "on your own terms" may result in all kinds of shenanigans: obfuscation, digressing, changing the subject, etc.
