When a bus driver or school mentions they will be driving a mud route that day the meaning is basically that the bus driver will be driving down an alternate route when their normal routes may get hazardous to drive on such as being muddy "hence the name mud route".
The mud routes are used when the bus driver has to drive a rural route on possibly dirt roads that can become muddy, washed out etc.
Rural school districts usually setup mud routes and on nice days or when the roads are in good shape the dirt roads will usually be driven down to pickup and drop off the kids.
But when the weather turns bad and the roads get bad then the buses will go on the mud routes which are usually paved.
Then the parents will take their kids to the spot on the mud route to have the bus pick up their kids and then they will meet at a designated stop where the parents will pick up the kids on the mud route after school.