Multiplier effect and GDP The fundamental national income accounting identity is: $Y = C + I + G + X - M$. The multiplier effect tells us that an autonomous injection into the economy will result in a multiplied increase in the real national income due to rounds of induced consumption. My question is, if that is indeed the case, then wouldn't the national income identity need to be modified to include the multiplier? That is to say, wouldn't it become: $Y = k_c C + k_i I + k_g G + k_x X - k_m M$?
You can't apply it to the national income identity directly because its components, namely consumption, investment and imports are functions of national income (output) and hence, they are not autonomous. In fact, in simple undergraduate level model, following Blanchard et al:

$$ Y = C(Y-T) + I(Y,i) + G  + X(Y^*, \epsilon - M(Y, \epsilon)/ \epsilon  $$ 

Where in addition to variables already described in your post $T$ are taxes, $i$ is interest rate $Y^*$ foreign output and $\epsilon$ is the exchange rate.

So you cant simply apply the multiplayer just to the variables itself as you would apply them to output multiple times. Rather you first need to solve the equation above for output. To do that I will assume two simplifications - $I$ will be fixed (that is $I = \bar{I}$ and consequently $I$ is no longer function of $Y$, and autraky to get rid of $X$ and $M$ - this is to simplify the math as now only $C$ is a function of $Y$ making solution easier (not doing so would only cause multiplier to be more complex and derivations longer). Moreover, assuming that $C$ is given by $C = c_0 + c_1(Y-T)$, the solution to the goods market equilibrium will be given by:

$$Y  = \frac{1}{1-c_1} \left( c_0 + \bar{I} + G - c_1T\right)$$

where the term $\frac{1}{1-c_1}$ is the multiplier (which you denote by $k$). Applying multiplier makes sense only at this step because only now Y was eliminated from the right hand side of the equation. Previously, you were applying the same multiplier to Y over and over again without noticing it. 

Also this is completely correct:


  multiplier effect tells us that an autonomous injection into the economy will result in a multiplied increase in the real national income due to rounds of induced consumption.


But no part of the original accounting formula contains autonomous spending aside form $G$. In fact the solution to the goods market equilibrium given by the equation where we eliminate $Y$ from right hand side gives you effect of autonomous injections as in that solution $c_0$ is the autonomous consumption, $I$ was assumed away to be autonomous by assuming its fixed (admittedly that was bit sloppy from me but it simplifies math and shortens the explanations) and $G$ and $T$ are arguably autonomous by themselves.