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Are you sure you understood them correctly? Why can the PSU (power supply unit) not be replaced? PSU's are generally easy to replace and these days not very expensive ... depending on the wattage required such as 400 watt, 500 watt, etc. With today's equipment requirements, I would not suggest anything less than a 400 watt PSU on even the basic of computers.
The processor (CPU) is a completely different component than the PSU, but, it is possible that the failing PSU could have caused damage to the CPU. When a PSU starts failing (which could sometimes take days or months), the regulated power supplied to the individual components could become less than is required, or more than required, both of which can cause damage.
So, is the CPU (processor) damaged or have you confussed what they said? If the CPU is actually damaged, chances are that parts of the motherboard could also be damaged. It may be time for a new computer.
It's not so much whether the CPU can read the hard drive or not. It's more the specifications of the motherboard (mb) used. If the current motherboard is not damaged, then there is no concern. If a new motherboard is required, then be sure the new mb supports your current hard drive ... ata, sata, etc.
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