Mud is a perpetual problem in many areas of the world. Draining a mud puddle is a big help. If the puddle is large or a river is near, use some sort of runoff control . In this case, if there is standing water, let it dry or drain it with buckets, shovels or whatever, then dig the mud out, leaving a hole of firm dirt. Fill the hole in with good dirt and pack it as firmly as you can.
Erosion is caused by water, wind and traffic.
Wind erosion is slowed by wind barriers such as trees and shrubs. Water erosion occurs from rain drops striking exposed dirt and from water flowing along exposed dirt. A canopy of leaves significantly reduces rain drop erosion. Flowing water erosion is reduced by getting the flowing water off the trail as quickly as possible. On a long downhill section of trail, frequent water bars to detour water off the trail help.