Can a bad control problem affect endogeneity? Suppose the population regression function is as follows:
$$y=\beta_{0}+\beta_{1}x_{1}+\epsilon$$
In this case, the assumptions of the linear model for obtain unbiased and consistent estimates are satisfied, namely that $E[\epsilon| x_{1}]$=0.
However, we mistakenly estimate instead
$$y=\gamma_{0}+\gamma_{1}x_{1}+\gamma_{2}x_{2}+\epsilon$$
wherein $x_{2}$ itself is irrelevant in a causal sense, and is itself
an outcome of $x_{1}$(the so-called bad control problem). Is it possible
then, that the conditional mean independence fails now? In other words,
is it possible that $E[\epsilon|x_{1},x_{2}]\neq0$ even though $E[\epsilon|x_{1}]=0?$
In other words, have we introduce endogeneity by including an irrelevant
variable? If so, why? Wouldn't that correlation with the error term
already exist?

We can find an example such that $E(\epsilon|x_1) = 0$ but $E(\epsilon|x_1,x_2) \ne 0$.
Suppose that $(\epsilon,x_1,x_2)$ take the following values with the same probability of $1/4$
$$
(-1,0,0), \; (1,0,1), \; (-1,1,0), \; (1,1,1)
$$
We have that $E(\epsilon |x_1) = 0$ since
$$E(\epsilon | x_1 = 1) = E(\epsilon | x_1 = 0) = -1 \cdot \frac{1}{2} + 1\cdot\frac{1}{2} = 0,$$
but $E(\epsilon|x_1,x_2) \ne 0$ because for example
$$ E(\epsilon|x_1 = 0,x_2 = 0) = -1 $$
