Work must now be done to move the electron sideways, as well as in the direction of the anode.
What does this extra work?
I'm not sure what you are asking here. Work is done by the second electric field, since the field will be parallel to some component of the electron's motion, and hence the speed and kinetic energy of the electron is changed by it.
Are you under the impression that there is some other work also being done (in addition to the work done by the two electric fields)?