Based on your description the executable you're trying to run, needs privileges that are not granted to the account you're using to run the service. As this is part of Windows security system, you can't change the behaviour.
For example if the executable to run needs access to a network path and you're running the service using a local account, then the started executable uses the local account and necessarily cannot access the network resources.
If you don't want to change the service account then try starting the executable using different, explicitly specified credentials. You can define the account to use by defining them in
ProcessStartInfo Class (System.Diagnostics)[
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