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Choice C is correct. Subtracting the expression $\left(x-29\right)$ from both sides of the given equation yields $0=\left(x-a\right)\left(x-29\right)-\left(x-29\right)$, which can be rewritten as $0=\left(x-a\right)\left(x-29\right)+\left(-1\right)\left(x-29\right)$. Since the two terms on the right-hand side of this equation have a common factor of $\left(x-29\right)$, it can be rewritten as $0=\left(x-29\right)\left(x-a+\left(-1\right)\right)$, or $0=\left(x-29\right)\left(x-a-1\right)$. Since $x-a-1$ is equivalent to $x-\left(a+1\right)$, the equation $0=\left(x-29\right)\left(x-a-1\right)$ can be rewritten as $0=\left(x-29\right)\left(x-\left(a+1\right)\right)$. By the zero product property, it follows that $x-29=0$ or $x-\left(a+1\right)=0$. Adding $29$ to both sides of the equation $x-29=0$ yields $x=29$. Adding $a+1$ to both sides of the equation $x-\left(a+1\right)=0$ yields $x=a+1$. Therefore, the two solutions to the given equation are $29$ and $a+1$. Thus, only $a+1$ and $29$, not $a$, are solutions to the given equation.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.