My biggest problem was that it was difficult to stay involved in the story when White kept interrupting to tell us he was skipping over a section and that we should read Malory's version, or slipping into archaic writing, or even directly quoting Malory's "Morte d'Arthur." It was almost as if he just couldn't be bothered writing his own version, and it makes the novel at times feel less like a story and more like a summary and critique of Malory's version.
Another thing which prevented it from feeling like a story you can be involved in is... pretty much every scene Merlyn (to use White's spelling) appeared in. The second volume, "The Witch in the Wood", and the fifth "The Book of Merlyn" consist almost entirely of Socratic-style dialogues where Merlyn tells Arthur about the dangers of war... we read these instead of an actual story about events where we can see these dangers for ourselves. This makes the book feel more like a collection of essays than a narrative. Merlyn also flattens out the story by telling Arthur about everything that's going to happen in the future, making it feel like the characters are just going through the motions of what they know they're supposed to do, instead of making their own decisions.
The reason why Merlyn knows the future, incidentally, is apparently because he lives backwards instead of forwards through time. Well, he sure seems to be living forwards whenever we see him. For one thing, he's good at speaking forwards. He can also remember the past fairly well, such as when he meets Arthur as an old man and remembers tutoring him as a young boy.
This is a fairly subjective criticism, but White's writing style often jars with the tone of the story. Both the narration and dialogue can sound quite patronising, as if White was underestimating his readers' intelligence (take a shot every time a complex moral or political dilemma is described as a "muddle"!). He also slips into a quite inappropriate whimsy at times... in the first volume, "The Sword in the Stone", Merlyn turns the boy Arthur into different animals so he can observe the way their societies work. So we get some interesting character archetypes with hunting hawks and so forth, and then... a bepectacled badger with quaint china-cabinets and leather-bound books, and a cockney hedgehog who calls himself a "tiggy". Does the spirit of Beatrix Potter really belong in an Arthurian epic?
White's cultural perspective could also lead to some problems: he compares knights to individual cricketers from White's own time, as if he expected the readers to be familiar with them, and makes other cricketing analogies, but I guess that's who he was writing for. Despite the fact that most of "The Book of Merlyn" is spent with Merlyn ranting about how people should stop identifying themselves with nations and instead think of themselves as individuals, there are some face-palming stereotypes in there. For example, Arthur folds his arms into his sleeves "like a Chinaman" and looks out across his country "from Zummerzet to Och-aye." Did White really just call Scotland "Och-aye"?
There are moments, however, when the story shines through the style, and when White writes enough dialogue and actions in contemporary speech that we can be involved in a story we can understand, and with characters whose thought processes relate to interesting and dramatic events that are actually happening within the story. A retelling of the Arthurian legend in language which modern readers can engage with is certainly something that should exist. I think it can be done a lot better than this, though.