As promised, I asked this question to someone more knowledgeable, and here is the answer:
I taught math for 6 years in inner-city schools and definitely feel your pain! IMO the issues are systemic and way outside the scope or control of the individual teacher. All you can do is to do what you must to protect your career interests while giving the best instruction you can to your students. It's unfortunate that this presents a conflict but sometimes that's the reality.
Depending on how strict your environment is, you may be able to spend a little more time where it's needed and find something to skip later, and/or teach the pre-skills that the kids don't know while aiming for the assigned topic. If anyone asks you are "addressing the common core standards" while "using a spiral curriculum approach to reinforce pre-skills," "differentiating your instruction," "providing accommodations to support all learners in your classroom," etc..
Kids who have struggled with math often assume that they will never do anything correctly and have shut down. However my experience is that even if they're putting up a front of apathy, they do care and appreciate chances to be successful. They benefit from lots of small opportunities to engage, learn, and succeed. I've found these strategies helpful:
Try a daily warm-up problem that focuses on pre-skills for that day's lessons or on a past topic that they didn't get. Give them credit for completing this.
Break procedures into small steps and give written descriptions of each in as simple language as possible. I like to write descriptions off to the side right next to each step of the math.
You can spend lots of time on sub steps and even if you cannot reach full mastery for all students in the allotted time frame, you are teaching what you were told to teach and they are learning something. For example, if your topic is solving quadratic equations, you might have mini-lessons on any of the following:
Writing factor lists of the constant terms. So they're practicing factoring of numbers and multiplication. Find factor pairs that combine to the middle term (for leading coefficients of 1). Now they're practicing combining negative and positive numbers. Give them the factorizations and have them set each factor to zero and solve. Now they're working on solving two step equations.
Try to talk for no more than 5 minutes at a time as much as possible. Do one or two steps of an example, and then have them repeat those steps on a parallel problem. You can walk around and give some individual attention to those who need it. (This is a "formative assessment" btw.)
You can also put them in small groups so you can help a few kids at a time. You might want to carry around a small white board for this. You can try "grouping heterogeneously" by grouping a kid who gets it with one or two who don't.
Try writing and distributing your your notes in advance but leave blanks for them to fill in. This helps them tune in as they'e looking for what goes in each blank. Again, I would award credit.
When you ask questions, start with really simple requests that you know for sure the particular kid can handle. i.e. simple arithmetic or reading the next step in the notes out loud. Give enthusiastic praise or thanks. What you're really acknowledging is the kid's contribution and engagement. And you can build to harder questions as the kids' trust and confidence grows.
Another way to minimize lecture time is activities that aim to have them discover properties for themselves. For instance, have them solve some problems like $x^3*x^4$ by expanding out and counting the x's. Hopefully after their hand cramps up from all the writing they'll figure out the shortcut!
Make two versions of each assessment by changing the numbers in the problems and give one as practice the day before. This way the kids will know exactly what to expect and what they will need to do.
You may want to give them a quiz on each topic that provides written instructions for each step. Then even if they're not quite getting the big picture, they are learning some of the pre-skills necessary and getting rewarded for this.
Give credit for lots of little things just by walking around and checking, maybe have a daily class participation grade. Don't take every little thing home or you'll drive yourself nuts. Kids like this kind of instantaneous reward anyway.
Hope this helps!!!