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When writing a program from the debugger to the MCU, is there a check (a process to confirm that the program written to the MCU is correct) in the program writing itself?

In things like Keil there is a debugger/flasher option to "Verify After Download"

You can also have a post-link step/method to add a checksum or CRC to the firmware image that the code in the firmware itself can subsequently recompute and validate.