Evaluating a minimum mean square error explicitly

Question

I have a minimum mean squared error (MMSE) expression that I would like to evaluate explicitly: $$ \mathbb{E}[(X+\mathbb{E}[X|CX+G])^2], $$ where:

• $X\sim\text{Bernoulli}(\nu)$ where $\nu$ is some fixed constant and $G\sim N(0,1)$;
• $X$ and $G$ are independent of each other;
• $C$ is just some fixed constant (non-random).

Question: Is it possible to evaluate the MMSE expression explicitly? Otherwise, how far can we go? I don't mind the integrations. Thanks.

Remark: This paper looks at what the form would take if $X$ is Gaussian.