I will finish a lawn that I am mowing if it starts to rain after I started and it is a light rain. One of the complaints I often hear from customers is that they did not like that their old LCO would mow in the rain and damage their lawn.
I always wait for the lawns to dry in the morning to prevent leaving clumps all over.
I schedule mowing Monday through Thursday. Friday is for rain delays and rainouts. No rain outs then Friday's are for up sell work.