Nor would it explain how he would copy it off Matthew and move it to the wrong place, unless it was an error. If it was an error, it doesn't seem like it is any more or less likely than if Mark wrote it rather than copied it.
Yes, either scenario is possible, though to me, Matthew's wording seems more likely to be original, because I would be more likely to say they began to pluck grain than to say that they began to make their way without qualifying where to.
This can be confirmed by comparing the whole paragraph with Luke's version of it. When Matthew and Mark disagree as to the wording of a phrase, Luke and Mark are in agreement. And when Luke and Mark disagree, Matthew and Mark are in agreement. If you're ready for a brain-bending challenge, read on:
And it came to pass (Mk)
And it came to pass (Lk)
At that time (Mat)He is passing through (Mk)
He is passing along through (Lk)
Jesus went through (Mat)the grainfields (Mk)
the grainfields (Mat)
grainfields (Lk)on the Sabbaths (Mk)
on the Sabbaths (Mat)
on a Sabbath (Lk)and His disciples (Mk)
and His disciples (Mat)
and His disciples (Lk)
were hungry (Mat)
began to make their way (Mk)
and they began (Mat)
plucking (Mk)
to pluck (Mat)
were plucking (Lk)
and eating (Lk)
the heads of grain (Mk)
the heads of grain (Mat)
the heads of grain (Lk)
and to eat them (Mat)
rubbing them in their hands (Lk)
And the Pharisees, having seen, said to Him (Mat)
And the Pharisees were saying to Him, (Mk)
But some of the Pharisees said (Lk) Behold, why do they (Mk)
Behold, Your disciples are doing (Mat)
Why are you doing (Lk)that which is unlawful on the Sabbaths? (Mk)
that which is not lawful on the Sabbaths? (Lk)
what it is not lawful to do on Sabbath (Mat)The above comparison shows that if Matthew and Luke are secondary to Mark, then whenever either Matthew or Luke decided to change Mark's wording significantly (seven times, by my count), the other followed Mark's wording closely. It's as if Matthew and Luke collaborated so that at least one of them would be copying Mark at all times (except when they added their own detail or omitted detail).
That could be coincidental; my suggestion, an extension of the 'Griesbach hypothesis' and proposed by Harold Riley, is that Luke had used Matthew and made changes to his wording, then Mark used Matthew and Luke, and conflated their wording.
As to how 'began' was moved to the wrong place: Riley thinks that Mark copied 'began' from Matthew, and added 'making their way' which is the same sort of action as 'passing through' and seems to be a tautology, possibly for dramatic effect. In doing so, 'began' became attached to 'to make their way' rather than 'to pluck'.