Do you include income earned but not received in calculation of GDP using the income approach?
No, because that income is only theoretical at this point. You can see in the definition of each part of the income expression of GDP that it is the amount received, because this is the amount that will actually be spent or saved, and we are after all trying to match the other definition of GDP which is total amount of money spent or saved.
In practice, this can get quite complicated. For example, we have to decide whether to count something like a defined-benefit pension (e.g. Social Security) as income "received" or simply income "earned". If you had actually gotten the cash and put it into a savings or stock account, it would be considered income and counted toward GDP. But as a defined-benefit pension, the money you are being forced to save is in a sense not actually there, and won't be until you get it. In practice, we count the "employer contribution" to Social Security as income, but not the net present value of future pension payments due to that contribution. Then when you actually get paid Social Security benefits years later, we count those payments as income. Are we double-counting? Sort of. But that's how we do it.
My personal opinion is that saving in general is misunderstood and overrated. When money is put into an account instead of actually spent on things, it's really not serving its function as money. Yes, in theory money that is saved corresponds to money that is invested; but that's only true if you use a really weird concept of "investment" that includes inventory accumulation, i.e. products piling up on shelves. In reality all real investment is itself spending, just spending on capital goods instead of consumption goods. I think our system of national accounts should be adjusted accordingly. But as they stand, our national accounts include saving as GDP, and therefore need to account for income that is saved.