I am wondering if the application of General Relativity is ever needed(?) I assume that Newtonian mechanics is sufficient with some trajectory corrections applied as needed on route. I also assume that the latter corrections must be minimal to save fuel, hence the trajectories are planned in advance as best as possible.
It depends on the precision you want to have.
If you want to do precise navigation, you will have to consider all relativistics effects, especially on the clock. There are general relativistic models for space-time coordinates, to consider dilatation of time (TDB scale time for example). You can find IERS conventions here. But this case only applies if there is a clock on board the satellite (like GPS).
If you want to just perform the navigation by considering accelerations you can choose to consider or not relativistic accelerations. The amplitude of the relativistic acceleration is close to 10-9 lower than GM central acceleration. There is this book (Moyer) which describes precisely accelerations to consider, particularly the point mass Newtonian accelerations and relativistic accelerations.