Given that this is low-frequency AC (50 or 60 Hertz), you can consider it to be DC for the purposes of impedance calculations on the PCB. The wavelength of these AC signals is much longer than the size of your entire PCB - about 5000 kilometers in fact. Removing the ground plane will have no effect on the propagation of these signals.
What it will have an influence on, though, is the path that return currents will take, as you already identified yourself. Given that these are power traces, this is actually quite important both to prevent unwanted resistive voltage drops and for EMI reasons. An unbroken ground plane will provide a better (shorter and lower resistance) return path, so it's always a good idea to keep the ground plane intact underneath power traces.
In your case, you should fill the entire bottom layer of the board with ground plane wherever possible.
What you might want to do, though, is to increase the clearance between the ground plane and adjacent signals on the same layer in order to prevent accidental bridges during soldering. This can be done in the design rule settings of the CAD software you're using (KiCAD?). 0.5mm to 1mm is probably a good idea. Another thing to consider is to increase the default thickness of your signal traces - they look awfully thin and there doesn't seem to be any reason to keep them that way (lots of empty space around them). 0.5mm is likely a good starting point as well there.
You could also consider flooding the top layer with ground plane too and stitching it to the bottom plane with vias in a regular-ish grid to lower the plane resistance further.
And lastly, it looks like you've placed some (ground) vias within the pads of components. This will cause you trouble with manufacturing unless you very specifically know what you're doing there. Don't do it with this board, there's enough space to run a short trace to a via slightly away from the pad. It'll save you a lot of headaches during soldering (via-in-pad can suck solder away if not done properly).
