pull through (phrasal verb)

You seem to understand it. "Make it" is similar. Let's say a vet is treating a dog, performing surgery on a dog. She says to an associate It was touch and go for a while (I thought the dog might die), but I think she's going to pull through/make it (survive).

One thing I'd add just to make it explicit - to 'pull through' can be standalone (and usually is). You don't need to pull through anything in particular. So your second example you don't need the 'this midweek' part, you can just say you've pulled through.

You are using "pull through" correctly in the first example. However, I would probably say "I pulled through this month" instead of "I've". I don't have the means to explain why, but I would say in normal speech we tend to simplify when we can." I am based in the US btw. For the second sentence, it sounds a bit less natural. I think it's because "pulling through" is something that occurs over time, not at a specific time. You don't pull through on a Wednesday. You pull through over the course of a Wednesday. Sublte difference, but it's important. Also, the term is "came down with a cold", and you don't hear the term "this midweek" (just say "midweek"). I would say "I came down with a cold last Friday, but I pulled through." or "I have since pulled through." If you wanted to associate it with a specific time, you'd say something like "I've been feeling better since Wednesday". Hope this helps, lmk if you need me to explain anything better.