Lf: there is stuff that can be put on the lawn to deter dogs, from chemicals to vinegar to motion activated sprinklers (I don’t think you can use it if your kids walk there). Not everything works, or can harm lawns, so do your Google research as best you can.
I think the best/simplest idea I saw mentioned (credit havahart) that might work for you is to put up a sign (with landlord’s permission) to the effect that the lawn has been treated (article mentioned against raccoons and skunks, in NYC it makes more sense to write against rats) so please keep dogs/cats off it. And then I’d go ahead and put some cayenne pepper or something harmless so it shouldn’t be a lie.
Good luck!