Warning: things are about to get a little weird. Here's the thing. Henry is the subject of this poem but he's also the speaker. We know. It's confusing. Just give us a paragraph or two and we'll try to clear this thing up.
Berryman wrote The Dream Songs (remember, there are a bunch of these things) as a kind of extended dramatic monologue. The primary speaker in the songs is Henry. The tricky part is, sometimes Henry refers to himself in the first person, and sometimes in the third person (as is the case in #29). Here's how Berryman describes the deal with Henry: "[The Dream Songs are] essentially about an imaginary character (not the poet, not me) named Henry, a white American in early middle age […] who has suffered an irreversible loss and talks about himself sometimes in the first person, sometimes in the third, sometimes even in the second." So, in "Dream Song 29" we have our speaker, Henry, telling us about… Henry.
It is almost as if our speaker is having a hard time telling the difference between Henry in reality and an imagined or a dream-state Henry. Sometimes they are the same guy; sometimes they aren't. The speaker even sounds confused, the way he mixes up the syntax in some of his sentences. He seems to be struggling to keep things together, to keep things in order. What we know for sure is that Henry is sad, sleepless, and prone to violent fantasies—maybe not the best choice for a lab partner.
It is always a good idea to try to separate the speaker from the poet, and Berryman wanted to make sure this separation was clear. But there are many similarities between speaker-Henry and poet-John. That said, Berryman insisted he was not Henry. He was quoted in interviews saying things like, "Henry does resemble me, and I resemble Henry; but on the other hand I am not Henry," and "Henry is accused of being me and I am accused of being Henry and I deny it and nobody believes me." Still, an artist can't help putting some of himself into his work. Henry is certainly from Berryman's imagination and, therefore shares some of his issues and anxieties. So, Henry is the speaker. Henry is Henry. And Henry is (isn't) Berryman. Got it? That's okay. Let it sink in for a while. You'll be okay. (Henry probably won't).