"We have to make a claim for the delay."

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I have a question about the phrase "make a claim".
I think it means "complain", but in the book that I found this phrase, it is translated as "ask for compensation".
Does "make a claim" also mean "ask for compensation" or "claim damages" ?

I suspect the book's translation is not very accurate. The book was written by a Japanese author, and I've found several mistakes, so I suspect this is another loose translation.

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"To make a claim," is a formal or an official assertion (often used by legal or formal institutes). Though the phrase can be used for formal "complaints," it doesn't always have to be. For example, look at the sentence, "I have to make a claim for my lottery ticket." In this example, it isn't a complaint, but a formal assertion (or will make that assertion) that you are the owner of the lottery ticket.

In general, "make a claim" can mean many things depending on context, but they will all be formal or official assertions.

With your example sentence earlier, "We have to make a claim for the delay," has a very different meaning to "We complained about the delay." I'm not sure which one you meant to say since there was little context. I can see people using the first sentence when traveling by airplanes, flights got delayed, and they want to be reimbursed by their travel insurance. I can also see people using the second sentence with a late train and people vocalizing their frustrations.

I hope this helps.