Isn't it canon?
I thought Greagoir sent him there after the Blight, when mages started returning from the war, and Cullen grew super-paranoid (what if they learnt blood magic from the darkspawn???), and Greagoir decided to send him to a quiet village to guard the local Revered Mother, and Cullen spent there a couple of months. I was going to make something about his time there, and what happened, and how it helped him, but I don't have time for that anymore.
I also headcanon that Cullen requested to transfer him to Kirkwall. It doesn't really contradict the codex, because codex probably means an official paper from Greagoir, and I imagine Cullen just sent him a letter, "It's too quiet here, and I feel better when I'm busy. Please sent me somewhere where I can be too busy to think about what happened." And Greagoir said, "Ok, you go to Kirkwall, it's always busy there." And wrote to Meredith, "Hey, keep that guy as busy as you can!" And she thought, "Awesome, I have a ton of paperwork nobody wants to do! I hope that new guy has decent handwriting!"
Mages in Kinloch Hold didn't know Cullen went to Kirkwall after Greenfell, because why would anyone tell them? They probably overheard him talking about Greenfell when he was leaving, that's all. So when we hear them chatting about it a year later, it's because they think Cullen still serves there.
It's canon that the mages said he went there.
I don't know that it's canon that he actually went - what is canon for him during the period of time between Broken Circle and meeting him in Kirkwall seems pretty vague.
That said, I think it makes most sense to assume he went. He was at the very least not at Kinloch Hold during Witch Hunt, and given how much more stable he appears when we meet him in Kirkwall, I think the best explanation for where he was is that he was in Greenfell receiving... maybe not what we in the modern world would call therapy exactly, but surely templars being traumatized wouldn't be an unusual thing, and the chantry would have developed methods of helping them recover.
It explains the time gap, and why anyone thought Cullen might be in a state to be successful at a new assignment, and why Cullen appears have experienced some recovery (although of course still suffering trauma after-effects) by DA2.
It also explains why there was a rumor about him killing apprentices, as people have said, at least to the extent that it needs an explanation; it's really not that hard for a rumor to get started.
I discovered that about rumors yet again this week when my residents seemed disproportionately upset that I was going out of town for the day to attend a seminar. I had let them know I was going (they tend to get upset if the expected people aren't there, so I wanted to give them some warning) and they had somehow interpreted that as I was being transferred to Syracuse forever and they were never going to see me again. I had to reassure them that no, I would only be gone for the day. So yeah, rumors - especially in a fairly enclosed system with only so many people to gossip about - can get really wild really fast. 