You said:

I want [the second attempt] to wait for the first refresh API finish

You can save a reference to your Task and, if found, await it. If not found, then start the task:

actor Refresh {
    var task: Task<Void, Never>?

    func refresh(value: Int) async {
        try! await Task.sleep(nanoseconds: 100_000_000) // imitation API CALL
        print("Try to refresh: \(value)")
    }

    func mockCallAPI(value: Int) async {
        if let task = self.task {
            _ = await task.result
            return
        }

        task = Task {
            await refresh(value: value)
            task = nil
        }
    }
}

Apple showed example of this pattern in the code provided the with WWDC 2021 video, Protect mutable state with Swift actors (but this code is not on the web site; only provided in the Developer app). See How to prevent actor reentrancy resulting in duplicative requests?

Their example is more complicated (a pattern to avoid duplicate network requests from being initiated by some image cache/downloader), but the kernel of the idea is the same: Save and await the Task.

Why do they need to be in seperate tasks if you don't want them to run in parallel, await means the code will no progress any further than after the task completes, because of cooperative threading the thread that initiated it may be used to do something else, like process more using interaction, or other tasks, in fact because you have them in seperate task you are asking for them to be run in parallel, its possible the block that contains them will enter the very bringing and you will have another two task, in which case you need to wait on a result from them container task saying they have completed, and the code that entered the block checks this to continue any further.