Answer: Unfortunately, there is no effective way to cover surface roots. Some trees such as oaks can be killed by adding soil over their root zones, and other trees such as maples will simply grow more roots at the surface all over again.
As you have seen, hitting the roots with the mower is damaging to the mower and also can be damaging to the tree.
Generally, lawn grass will not grow well in shade, especially when in competition with surface rooted trees. Since you are already having trouble with the roots, you may want to consider switching from lawn to a groundcover type of planting such as Vinca minor.
Q&A Library Searching Tips