'Got the monk on' in northern England
Ian McMillan, Chelp and Chunter: How to Talk Tyke (2007) has this entry for the noun monk and the phrase have the monk on [combined snippets]:
monk noun have the monk on to sulk If tha's got the monk on again I'm stoppin' in
The same source uses the phrase in connection with other phrases as well: "Love to join you, lads, but our gert's got the monk on about summat" and "Oh that mardy git. He's allus got the monk on abaht summat." (For the meaning of Tyke, see Why is the Yorkshire dialect called 'Tyke'?)
A search for the phrase turns up several fairly recent publications that use it precisely in the sense of "sulking." From Ajay Close, Forspoken (1998) [combined snippets]:
'Don't be so nesh.'
Sometimes she says I'm mardy or observes that Drew's got the monk on or, nipping out to the newsagents, she'll ask if I want any spice? And I don't think she knows how my heart trips because I hardly understand it myself, the persistent sweetness of words I never used. Sometimes I think her vowels have become even more mushy pea since she left England. Only an accent so artificial could survive seventeen years in America intact.
From David Waddington, Out of the Ashes?: The Social Impact of Industrial Contraction and Regeneration on Britain's Mining Communities (2001):
We both enjoy each other's company. We have terrible rows, which is stress to me. We've got a good relationship where we can talk things out. It does get a bit stressful at times when we're going through a phase where we've both got the monk on with each other, but it does boil down to we've got each other to talk to.
And from Richard Cameron, Gong Donkeys (2004):
Robert Got the monk on, 'as she? (To David, by way of explanation.) Stands at the gate, watchin' stars. Or 'as she gone to bed? Pickled onion.
He goes back through into the kitchen, humming a song to himself. It might be 'Food Glorious Food'.
But in another instance it seems to refer to conduct that's a bit too kinetic to fall within even a broad definition of "sulking." From Derrick Allsop, Fighting Chance: Winners and Losers in the Ultimate Risk Business (2011), Martin Jolley speaking:
'I was in a pub in Chesterfield and because they knew me, I got served before this bloke who as at the bar. I went back to my seat but he got the monk on and the next thing I knew a glass was smashed on my head and blood was pouring down my face. I lost three pints of blood. ...
'If I was ever like Ali, I know I've got friends who would help me take my life. I wouldn't want to be a burden in anyone. You see the medical checks we have before each fight. They weigh you again to make sure you're not dehydrated. ... I'm probably 12 stone 10 lb now. But I wasn't going to get weighed tonight until we sorted out with the money. They knew better than to try and mess with me. They could see that I'd got the monk on so when I told them to get stuffed that was it. Nobody was going to argue with me. ...'
As Jolley uses the expression, "got the monk on" means something in the range between seething with anger and violently enraged.
'Get [one's] monkey up' elsewhere in England
This in turn suggests a connection between "got the monk on" and the following allied expressions listed in John Ayto, Oxford Dictionary of English Idioms, third edition (2009):
have (or get) your monkey up be angry
put (or get) someone's monkey up make someone angry. informal
This expression is far older, with mentions of it going back almost a century. From John Russell, "Jonah" in The Saturday Evening Post (September 15, 1917):
He moved a pad-foot nearer, and Gunny winced and shrank.
"Oh, you needn't get your monkey up. I ain't said no—'ave I? Didn't I take your side of it when the rest was crazy to chuck the cruise? I wouldn't stand in when they wanted to scrob you."
From Effie Albanesi, Fine Feathers (1928) [combined snippets]:
"I heard he had telephoned through to mother to say he would be kept rather late, and she wasn't to wait supper. I think I'd better keep out of his way, don't you? I don't want to have him jawing me and telling me I'm a murderer or something of that sort. And, I say, Gwennie is out to-night too. Joe rang her up and she's gone to meet him and have a little dinner and go to a theatre. Isn't it about time you got some one to trot you round, Polly? Oh, all right, don't get your monkey up. What a spitfire you are! Can't speak to you these days. Oh, must you go?"
From Myrtle Johnston, The Maiden (1932) [combined snippets]:
"Well I'm sorry. I didn't mean to get your monkey up. You were saying something weren't you——"
"I was saying it don't need more than a ha'porth of spunk to see things in this ship goes the right way." He leaned forward suddenly and his voice, dropping lower, lost its flippancy.
And from The Linguist, volumes 27–28 (1965) [combined snippets]:
The difference between a human child and an infant monkey is the potential ability of the human child to talk. We need not add "human" to "child." Only the young of the human race are called "children." There seems no specific word for the young of monkeys, so we call them infant or baby monkeys, and in return we sometimes call mischievous children "young monkeys." Colloquially, "to get your monkey up" is to get in a temper.
The expression "got the monk on" seems peculiar to the northern part of England. As the poster says, the primary meaning seems to be "sulking," but it can at times cross over into a form of active anger. This raises the possibility that the expression grew out of the older expression "get [one's] monkey up," meaning "to be angered." In fact, Eric Partridge, A Dictionary of Slang and Colloquial English, eighth edition (1984) says that "monkey up" goes back to the first half of the nineteenth century:
monkey up. A C.19 ... coll. term applied to anger and temper. At first in phrase, as to have (one's) monkey up, to be angry, and to put (someone's) monkey up, to make that person angry (both in Benj. Webster, The Golden Farmer, 1833). ...
A lineal connection between "get one's monkey up" and "got one's monk on" is circumstantially plausible, but I haven't found any authority that offers to confirm it.