I have a personal response to the question of weak spots, and also several more general responses. I'll begin with the personal.
I'm first and foremost a musician, so I perceive musical qualities in poetry as being in the foreground. This means that the musicality of the line is often very present in my awareness. As a result, I often cannot hear classical meter or rhyme without them getting rapidly sing-songy, which just irritates me to distraction. de DAH de DAH de DAH de DAH ed DAH — bleah! Drives me up a wall. I tend to hear most unsubtly metered and end-rhymed poetry as either imitation-music, or bad music, or lyrics in search of a song. There are numerous poets out there who have no musicality whatsoever, that's true. But there are just as many neo-formalist zealots whose ideas of musicality are ham-fisted and crude: they bludgeon you with end-rhyme and meter, they use unnecessary adjectives, just to force the line to fit the form, and so forth.
Here's the truth: all poetry, even prose-poems, contains rhythm and rhyme, melody and phrase: of all literary forms, the language of poetry is surely the most musical in nature. (Of course, great prose is also musical in its own way; defining a boundary between great poetry and great prose can be problematic.) But the elements of musicality can be used subtly: embedded internal rhymes, slant-rhyme and off-rhyme, alliteration; the rhythm of everyday speech, raised a level or two into musical speech. Sometimes that means finding a more musical syntax than the purely prose rules of grammars would allow. Modern free verse and blank verse have their own internal music, when done well.
I look to Shakespeare as an example of how to do this well. I have John Giedgud's readings of The Complete Sonnets (two CDs), and you can hear from Gielgud's readings how Shakespeare makes the sentences and pauses move across the rhyme and meter, so that often it sounds like high-flung prose. The end-rhymes become internal rhymes because the sentences are placed across the line. The first mistake most ham-fisted formalists make is to always end their phrases and sentences at the end of the line: this only exaggerates the end-rhyme. I suppose it takes a dramatist's ear for nuanced speech, to pull this off. So many imitate this, and fail. One of the only modern playwrights who could pull this off was Christopher Fry; his The Lady's Not for Burning is a masterful example. So for that matter is Edmond Rostand's Cyrano de Bergerac.
One of the only poets who can pull off sing-songy rhyme and meter without it becoming tedious is Longfellow. Mike Oldfield set a long section of Hiawatha to musical chant as one section of his large work Incantations, and it works beautifully. It brings into the foreground Longfellow's feel for musical phrase and rhythm, in a way that never gets tiresome.
As a poet, another weakness I admit to is that I have little feeling for the poetic epic. I am capable of getting immersed and never losing attention in a several-hundred-page-long novel. Yet long poetic epics can't hold my attention so well. The problem is often that novelistic discourses into description and philosophy, in a long epic narrative poem, can kill the epic's momentum. If you're going to attempt narrative, be propulsive. If you're going to write stream-of-consciousness, the epic poem may not be the best fit, in terms of form.
To hear long poems well, I go back to the source, the Homeric Hymns; Alfred North's study of epic song, The Singer of Tales, proved that Homer's epics were sung and chanted to music, not recited as voice-only text. If I have a limitation here, or a weak spot—and I'm not fully convinced that it is a weak spot—it's the feeling that most poetry works better when set to music. As every songwriter knows, you can get away with more, when the words are set to music.
So, if I have an overarching weak spot, it's that for me the musical line in a poem really makes it or breaks it.
Here's another weak area I see many contemporary poets share: they don't perform their poems well.
In my experience from numerous poetry readings over recent years, and poetry performances (and there is a difference), most poets are shoe-gazers who kill their own poetry when they read it aloud. That's because most poets read their poems in a flat monotone, with almost no sense of rhythm, or music (ahem), or performance. They know what the words in the poem are very well, having written them, and they tend to forget, perhaps out of nervousness, that the audience has never heard those words before. I find this to be true of all genres of poetry, formal or free. It's a problem of performance, and it reflects the contemporary situation in which "spoken word" has become divorced from mainstream poetry.
I was in a band for ten years called Dangerous Odds, that featured two core poets and four or five core musicians, plus invited guests. The music was spontaneously improvised to accompany the poetry performances. Both of our core poets were like jazz players, they had a sense of timing in their declamation, and the lingering over a word or phrase to give it special meaning and attention. They would even repeat a line at times, because they knew the importance of dramatically conveying even the most abstract poem about particle physics or chaos theory required a dramatic performance.
Here's a recording of one of these poets, the late Dr. Larry Giles, reading his poem Clocks in Chaos from his book The Chaos Poems before a live audience; listen to how Larry phrases, pauses, turns the tempo up or down, at will.
When we had guest poets at live or radiobroadcast gigs, the contrast was immediately noticeable: most poets (with rare and wonderful exceptions) rushed through, breathlessly, without pacing or rhythm, as though they were nervously shouting prose into a hurricane. They read their poems too fast, usually, probably from nerves, and the musicians then never got a chance to develop any ideas very far; it's like coitus interruptus when that happens, very frustrating to the musicians, who never get to go very far with an idea before having to abandon it for something else. No matter what the theme or tone of the piece, these poets didn't respond to the music responding to them, creating that lovely feedback loop of performance intention, and let's use the word now—acting—that was required. By contrast, you can hear the band behind Larry in the piece above having the time to stretch out and really go with a musical idea, in tandem with the poet, not fighting against him: a true collaboration.
Most poets, even those poets who claim to read their work out loud to listen for rhythm and music, in my experience, still kill their poems when they read them aloud. We get too close to the words, and we care too much about the linguistic meanings of the words, and forget to be performers. Dylan Thomas was one of the rare exceptions to this near-universal practice; listening to him reading his work is spellbinding.
As a result, I've gone out of my way to try to read my own poems, when I do read them, with a sense of performance supporting them. Not that I'm very good at it, but at least I'm thinking about it. It gets back to that musicality thing.
Now, the question might be asked, can one read non-metered, non-rhymed poetry like this? Absolutely. (I have the recordings to prove it. Several CDs worth of Dangerous Odds, for starters.) Thinking that only formalist, metered, rhymed poetry can be musical—which is an assumption I think many poets fall into by default of ignorance—is not only erroneous, it's pernicious. It's like equating music only with popular song ballad forms, which is a very narrow, very limited window onto what music is, was, and can be.
Another thing one can learn from Shakespeare's Sonnets: to enjamb where it makes sense to enjamb, and to follow the form where necessary, but to let the breath and the sentence play against the enjambment when necessary. And, thereafter, to read the poem as heightened speech, and not arbitrarily make breath-stops at the end of every line unless the phrase or sentence specifically tellls you to. Look again how in the Sonnets, sentences spiral around the end-rhymes so that they become heard as internal rhymes, not sing-songy stop-rhymes.
So, I think it's a nearly universal weak spot among poets to stop short of considering the musical side of their poetry, especially when performing it aloud.