Consider the covariance of an MA(1) time series $Y_t=\epsilon_t-\theta\epsilon_{t-1}$ at $h=1$, where $\epsilon_t$ is a white noise term with mean $0$ and variance $\sigma^2$.
$$p_1 = Cov(Y_0,Y_1) = \mathbb E[(Y_0-\mu_0)(Y_1-\mu_1)]=\mathbb E[Y_0Y_1-\mu_0Y_1-\mu_1Y_0+\mu_0\mu_1]$$
By linearity of expectation we have:
$$Cov(Y_0,Y_1)=\mathbb E[Y_0Y_1]-\mathbb E[\mu_0Y_1]-\mathbb E[\mu_1Y_0]+\mathbb E[\mu_0\mu_1]$$
Now I want to know which terms are independent, if any?
I know that $Y_0$ and $Y_1$ are not independent and $\mathbb E[Y_0Y_1]=-\theta\sigma^2$ and that the expectation of the other terms are $0$.
So are the following terms independent and expectations like this? $$\mathbb E[\mu_0Y_1]=\mathbb E[\mu_0]\mathbb E[Y_1] \\ \mathbb E[\mu_1Y_0]=\mathbb E[\mu_1]\mathbb E[Y_0] \\ \mathbb E[\mu_0\mu_1]=\mathbb E[\mu_0]\mathbb E[\mu_1]$$
What you are trying is: $$E[\mu_0Y_1]=\mu_0E[Y_1],$$ $$E[\mu_1Y_0]=\mu_1E[Y_0],$$ $$E[\mu_0\mu_1]=\mu_0\mu_1,$$ since $\mu_0,\mu_1$ are constants (${Y_t}$ process is a stationary process implying constant mean).