No, the geodesic equation is the analog of $x''=0$ (N2L with no external force). With the presence of electromagnetism, the equations of motion become as you have stated and is the analog of $x''=F/m$ with a nonzero (electromagnetic) force. Now, while this is the equation of motion, it is not the geodesic equation. The geodesic equation is what a particle would do under the influence of no external force, that is it (locally) moves in a straight line. The geodesic equation is purely geometric, while the force law is more physical. With respect to the geometry, you can measure the acceleration of the particle moving, hence the "error in the geodesic" or "force" in the right side of the equation. You can think of the EM force as analogous to an external force of an SHO or other such classical system.
Now, you can "couple" to the Levi civita connection (i.e. add) the electromagnetic potential, which is mathematically a connection on an auxiliary $U(1)$ i.e. $\mathbb{C}$ bundle (technically, this is a connection on $TM\otimes L$ with $L$ the line bundle chosen, which is usually trivial). Taking the geodesic equation w.r.t. this coupled connection gives the force law of EM which you wrote down. So in a sense, it is a geodesic equation, but not the purely geometric (i.e. from the manifold itself) geodesic. This sorta corresponds to saying that if you keep twisting your local frame to counteract the electromagnetic force, you won't see any acceleration. Big words, but the summary is "kinda". More generally, these tricks give Yang Mills equations by coupling the Levi civita connection with other "gauge connections".