Constant term in action term in general relativity?

Question

So I recently pondered the following. Let’s say I have an $2$ actions $S_1$ and $S_2$ which differ by a constant:

$$ S_1(dot x_i, x_i) = S_2(dot x_i, x_i) + tilde c$$

Now their equations of motion will be identical in classical mechanics (without General Relativity) upon varying the coordinates $x_i to x_i + delta x_i$. Intuitively, I know this constant term will make a difference in general relativity. Is this hunch correct? What does the constant term $tilde c$ look like in the form of Einstein Field Equations?

$$ G^{mu nu}+ Lambda g^{mu nu}= frac{8 pi G}{c^4} T^{mu nu} $$

Or is there a better way to get the equations of motion in general relativity? Directly from the classical (without General Relativity) action?