Blind source separation on images - with a known source
You already know one of the sources here, so you don't need ICA.

There are infinitely many couples $k$ and $B$ that verify:
$$
A = B + kC,
$$
 but in your case if you assume that $B$ and $C$ are uncorrelated, we can choose the solution that minimizes the correlation between $B$ and $C$.

Using the previous equation and the linearity of the correlation operator, we have:
$$
\text{corr}(A,C) = \text{corr}(B,C) + k \text{corr}(C,C)
$$

So you need to find $k$ that minimizes $(\text{corr}(A,C) - k \text{corr}(C,C))^2$, which is simply:

$$
k = \frac{\text{corr}(A,C)}{\text{corr}(C,C)}.
$$
And finally:
$$
B = A- \frac{\text{corr}(A,C)}{\text{corr}(C,C)}C
$$