If i have a sum say
$$C\sum_{l=1}^C\left( a+b\cdot c_l\right)$$
Is is true in general that $$C\sum_{l=1}^C\left( a+b\cdot c_l\right) = a+\left( C\sum_{l=1}^C\left(b\cdot c_l \right) \right)$$ and $$C\sum_{l=1}^C\left( a+b\cdot c_l\right) = a+b\left( C\sum_{l=1}^C\left(c_l\right) \right)$$
Sorry if this has been asked before i wasn't able to find this specific identity.
EDIT: $$\frac{1}{C}\sum_{l=1}^C\left( a+b\cdot c_l\right) = a+\left( \frac{1}{C}\sum_{l=1}^C\left(b\cdot c_l \right) \right)\label{1}$$
What you wrote is true if you replace your first terms of $a$ with $C^2(a)$. This is because you can split your summation into $2$ parts, i.e.,
$$C\sum_{l=1}^C\left( a+b\cdot c_l\right) = C\sum_{l=1}^C a + C\sum_{l=1}^C\left(b\cdot c_l\right) \tag{1}\label{eq1A}$$
With the first term on the right side, since you are summing a constant of $a$ a total of $C$ times, you have
$$C\sum_{l=1}^C a = C(Ca) = (C^2)a \tag{2}\label{eq2A}$$