Why would mutation be considered by some as a difficult concept to grasp? Why do some instructors delay teaching mutation due to considering it to be a more difficult concept? (than functional or recursive concepts, etc.)

It is very likely that, back in the 8-bit PC days, many thousands of 8 to 12 year olds learned to code in Tiny Basic or other "street" BASIC implementations (where there weren't even local variables or other easy recursion or functional semantic support).  How could these kids do so if variable mutation was a difficult concept?

Or is the spaghetti code they often produce a result of not really understanding the concept itself?  (rather than just the software engineering downsides)
I personally teach mutation almost immediately.  However, I agree with Buffy that more difficult is not really the metric people are using to decide to teach mutation later.  It is about giving certain key habits a chance to grow and develop before you get there.

Most people find recursion naturally more difficult than a loop.  There is no reason why this must be, and folks who learned recursion first will often find the difficult one to be the standard imperative loop.  If you begin a student in a programming world without mutability, what habits will grow?

Certainly recursion will come quite naturally.  So will linked lists and trees.  And certain sloppy habits will be naturally avoided, such as this balderdash:

System.out.println("You found " + hotdogs + "hotdogs!");
System.out.println("How many glasses of water will you drink?");
hotdogs = myScan.nextInt();


Ultimately, there is a deeper philosophical statement being made by the instructor about which habits they would like to inculcate and foster at the very root of their students' thinking.  These patterns of thought, it is presumed, will modify the initial ways that the students approach problems even later in life.  

I do not know of any research that backs this up, but, of course, much of what we try to impart to our students is not from research, but from intuition. And, who knows?  They might be right about this one.