Okay, I'm approaching this post assuming that your students already have adequate enough language skills for them to attempt essays and improve from there. I know in my case, my average was horrible but my language skills were above average; I just hated school.
So, in this post, instead of thinking of the 'problem student' as someone with poor English skills, we'll think of the 'problem student' as someone who lacks the motivation or, perhaps more particularly, self-confidence to fully engage with your lessons plan.
I do try to start discussions with them or ask them about what they think about the text but I still have to do most of the talking because they either give me 1-2 word answers or they just tell me that they don't know what to say.
I think your lesson plan is pretty good. Well structured and to the criteria, and I think it would work pretty well for the motivated student, but it's for the unmotivated student that it's just not applicable. There are at least three things that I think a 'problem student' would need to hear that a 'good student' wouldn't necessarily need to succeed:
1. Purpose
2. Relevance
2. Motivation (including reassurance, and including a reinvigoration of the English curriculum)
On #1 - how can you expect a student to engage with the historical context of the text when they don't know the purpose of the entire AoS, or the purpose of today's lesson
within that AoS?
Perhaps my entire post is only applicable to students of a lower socio-economic status seeing as those are the students I'm most familiar with but, I think a lot of the time with a 'problem student', before you answer their questions about themes and characters, you need to answer the quintessential question of the Cynical Student: "What's the fucking point?"
Get them excited; give them a purpose! (Note: this is pretty intertwined with #2)
Consider Language Analysis, which I think is the most relevant AoS. A lot of students that I've worked with (mostly students that went to my old school that hated their teachers while I was in Year 12

) would look at Language Analysis and think "wow, this is so shit... We're literally just doing the same thing over and over again and pointing out appeals to emotions or whatever". Of course that's dry and boring. No one in the whole entire cosmos could be interested in Language Analysis if their perception of it was so mundane; you need to show them the
purpose to Language Analysis, which will in turn show them the relevance of Language Analysis to their lives. In my experience, this will get them excited. (But I also speak with a lot of excitement so it's probably a mix of that as well).
Take a stereotypical gangster student from a shitty public school who still wants a 70 ATAR. It's easy to judge such a student and think "why does he even want an ATAR" but, this stereotype is common enough from what I've seen. Usually this kid will have at least one absolute c*** of a parent, they'll be "taken in" by students a few years older than they are so by the time they're in Year 12 they've already well-developed their persona, despite still wanting to do moderately well. Still, the purpose and relevance might be a struggle. And how would you ever give Language Analysis essays relevance to this student? Give him this example:
"You fucking dog!!:
Show him how 'dog' is a manipulative word. How is that trying to paint the recipient of the insult? Dogs are nice animals, so how have the connotations of this word made it a viable insult? Show him how people are trying to manipulate him everywhere he goes - this is something he understands, and something everyone will understand. Show him how parents, teachers, the government, the media use language to manipulate - how it is a
weapon. You could talk about body language - how splayed arms before a fight try to show confidence but really, the guy splaying his arms is just trying to manipulate you with language. And so, the purpose of Langauge Analysis is to be able to
identify how someone might use written or visual language in a way that could persuade a target audience .
This goes for all students, not just a 'gangster'. That was just one of the more difficult scenarios I could come up with, but with any other 'problem student', just think of something relevant to their lives. I use the example of asylum seeker propaganda pretty frequently - "boat people" or "illegals" versus "people who don't want to die". Mix that in with a "how dare the government try to manipulate us!" and you might have turned a 'problem student' into an 'indignant student' - which is ultimately a lot better for you.
If you show them that there the purpose goes beyond "getting the score out of it" and well into their lives for years to come, you might start to see some improvement.
It's not so clear cut for Section A and B. You'll have to get a bit more creative. Perhaps the purpose of Section A is to develop critical thinking skills. Perhaps it's argumentative skills. -- I often go the latter, because I think it's better to say to a student "try to dominate everyone else in an argument and show them that you're right" than it is to say "try to develop your critical thinking skills". Ultimately, the purpose is both up to you and up to them for the other two sections. (Whereas I suppose LangAnalysis is pretty clear cut).
Thus, part of my first lesson will be discussing exactly what I've discussed in this post. I will first go through the purpose, perhaps using the same aforementioned examples and generally just getting them to see the AoS in a new light, trying to get them excited. Then, every lesson after that, at the start of my lesson I'll say "okay, what do you want to do today" etc etc, try to get that sorted out quickly, then "Okay. Here's what we're going to do in today's lesson: xyz. The purpose of today's lesson to our overall purpose/AoS is: dfsdfg"
So 'purpose' is two-fold. Overall purpose, and purpose for every lesson after that. Relevance is the same and, as mentioned, follows very naturally from the purpose.
HOPEFULLY, motivation has stemmed from here and your troubles are over. If not, well... fuck. If after the first two a student is still dead to the world, there's most probably an underlying issue. Perhaps grammar and language troubles as mentioned by EZ, but perhaps crippling self-confidence problems (not that rare in my experience). This is where you need to be really empathetic and you'll have to coax work out of your student. I can't really advise you on this.. Try what I'll call "reflective empathy". Eg, try to feel what they feel, then reflect that. If a student thinks you'll judge them for their writing - "Man, when I was in high-school I was so fucking scared that people would judge me for my writing, so I swear to God there's no fucking way I'll ever criticise you, and if you ever feel really sensitive about a particular piece of writing, just let me know". Build trust, respect, admiration.
Most importantly, don't fall into the trap of dichotomising students into "good student" and "shit student". A shit student, more often than not, is someone who has been failed by the system or failed by their families. For so long I was a "shit student" - wagging class, definitely not paying attention while I was IN class, generally being outspoken about how much I hated the teacher etc. Some people thought I was on drugs because there were always massive rings under my eyes and my eyes were reddish from lack of sleep. It'd be so easy to look at me and think "man, what a shit student!!" but I had things going on which, to me at the time, were far more stressing than school could ever be. I'm not trying to take all the blame away from myself - I was still a prick - but there were also other factors. I've found since tutoring that it's so easy to dichotomise... I love teaching motivated students. It's easy. It's successful. It's fun. But a 'problem student' becomes much more challenging, difficult, and sometimes less successful. It's so easy to turn the frustration into animosity toward your student. Even with my own personal experience, I've STILL been tempted toward "omg such a shit student ffs!!!" but it's really important to fight the urge, and it might make your job easier if you start to think "they're such a shit student, I don't know what to do with them" to "there must be an underlying problem. I'll try to figure it out, then I might be able to figure out what to do with them".
Anyway. That's my spiel! Some good stuff in this thread as a whole for tutors, I think

. Good luck with your students everyone.