To keep on means to continue an action, just as “keep” can by itself. The difference is this: usually we use it to talk about a situation when you continue something in spite of some factor that could prevent you from doing so.
Examples: Katya kept on singing in spite of her brother’s protests.
I don’t know why she keeps on doing this to herself. You’d think she’d eventually learn.
Give up (don’t stop, don’t give up!)
To give up is to give up hope or to stop doing something because you don’t think it will work. You can also use it with a noun to mean getting rid of something you like.
Examples: I give up. I’m never going to learn calculus. I’m dropping the class.
He gave up smoking a year ago, and he’s so much healthier now.
For an extra (and extra-inspiring) example, check out this video:
“Go through” has a couple of meanings that may come in handy on the TOEFL. The first meaning is a synonym for “summarize” or “explain”.
Examples: The professor goes through the different amino acids and their properties.
Can we go through that one more time? I don’t understand.
The second meaning means to have an experience. This usage will be useful on a wide variety of independent speaking questions. For this usage, you can also say “be through”.
Examples: This is the most frustrating experience I’ve ever gone/been through.
I know what you’re going through, and I can help.
Have to do with
To have to do with something is to concern or be related to it.
Examples: What does this have to do with the topic we discussed in the last lecture?
This book has to do with a woman who is being punished for a crime she didn’t commit.
To make up also has two meanings. First, it can mean to make peace with someone after a disagreement.
Examples: Hug and make up, children.
You can lose good friends by not making up after superficial arguments.
The second use of “make up” is to comprise. It’s often used in the passive voice.
Examples: The book is made up of a series of vignettes.
These eight planets make up our solar system (poor Pluto).
Put off (procrastinate, deter)
If you put something off, you procrastinate.
Examples: If you put off studying for the test, you’ll have a bad time.
I put off writing this report, and now I have t o write the whole thing in one night.
If you put someone off of something, you deter them from it or cause them to dislike it.
Examples: Watching the process of making chicken nuggets will put you off them forever.
He was put off of basketball after he injured his knee in a championship.