Citrus trees won't like an under-crop. Look at the tree in its natural state; it forms a dense
canopy all the way down to the ground to exclude anything else from growing under it. It doesn't want an undercrop, it wants to be alone. Where citrus trees have been paired with other plants, like date trees, they
are the undercrop. If there is some extra space in between the citrus trees, you might want to consider adding some date palms.
Mangoes are a different thing entirely. They can be huge trees, 30-40m in height. There is lots that can go on under a mango tree, including putting a house under one. Being tropical trees, they have evolved in conditions of high temperature and humidity and are used to being exposed to lots of fungi. Unless you have your mulch a meter high, the mango tree is probably not going to notice some fungal activity going on at the base of the trunk. However, sprinklers have a tendency to bleach the trunk of a tree, and while that can look alarming, it's not doing the tree much harm. In your climate, the dry desert air is not going to let much in the way of fungi grow on the trunk of the tree. I would be more worried about the amount of salts in the irrigation water. Over time, the evaporation of water from the sprinklers may leave salts on the tree trunk, and that could potentially harm them.
