Interresting question.
C standard does not define access of a "carry bit", however when you write in C, the compiler will use the CPU's carry bit as needed if the target has one.
If you want to specifically use hardware directly such as a CPU carry bit, you must use assembly (i.e. inline asm).
As far as data sheets, the MCU data sheets do explain the carry bit, as they should. I wonder what data sheet you are referring to, but you don't have to answer.