I commend you handsomecat for trying to help your students. It appears that you are in a really unfortunate situation, since you are the only one who is searching for ways to help his/her students learn and have little control over the direction of the course. Still, you can learn much from this experience that will benefit your teaching practices in the future.
The students really seem to lack the prerequisite knowledge to be successful, which is a large reason that they expect to be spoon-fed answers. When your course first began, did they have this mindset? I would imagine that they came to class eager to learn, but soon after they became frustrated with their learning and developed this poor attitude.
You have a large challenge to meet with the hour time limit for each class. If I were in your position, I would try to shy away from teaching basic mathematics as much as possible to focus on the content of your course. It seems like you have a good general idea as to what mathematical ideas that your students are lacking in, so I would develop practice problem sets with a few review examples solved step-by-step and answers to each problem so that students can complete it outside of class as review. If they have difficulties, they can email you. Of course, this will take more time on your part, but would be beneficial for all and allow you to focus more on the course material itself. And if they don't want to devote the time to their own learning, it is with their own volition and you should not feel bad for this student who does not take advantage of this opportunity.
It also sounds like you do a lot of one-on-one mentoring during the tutorial session. This is fine but I would switch to more of cooperative approach with interactive activities. Here is an example. Since I am not familiar with Macaulay notation, let's assume that the students are trying to determine some quantity relating to the common problem with a block initially at rest on an incline. First, place the students in teams of two or more. Have them draw the diagram for the problem and then draw and label arrows for all forces present on the block (gravitational, normal, and even frictional forces). As they work together to solve this first task, visit each group and help as needed. Draw these forces on the board and review as needed. Next, have the students transfer these forces to a free-body diagram and determine the individual force components. Draw this on the board and review. Lastly, have the students apply the equations from their diagrams and solve for the given quantity. Do on board and review. This method has the benefit of breaking down the problem into small steps, keeping all the students on the same page, permitting students to help each other out so you don't have to do all the work, and allows you to give help to two or more individuals in the group rather than an individual student.
At the end of lesson, I would construct a mini-review. Prior to class you should make up four index cards for each student, each with an individual letter A,B,C, or D. Give the students multiple choice review questions that reflect the important points of your lesson and require all students to hold up their answer. This will permit you to see their grasp of the presented content during this hour, and what you may need to review upon for the next session. If you note major problems, you could email students more examples with descriptions to help with their homework. For example, I would give students four diagrams about the problem previously discussed, each with a different orientation of force vectors, and ask them to identify which diagram is correct. Another example would be to give students the diagram with the force vectors already drawn appropriately but four different choices for the free-body diagram.
