What is the white paneling under the vinyl on my house, and how can I fix it?
Given that someone put a gash in it with a string trimmer, its likely stucco over rigid foam insulation panels.

If there's wire mesh in there as well, it's likely a traditional multi-coat stucco finish that can be repaired using stucco repair techniques.

If there's no wire mesh, you may have an EIFS wall (exterior insulation finishing system), which is kind of insulation and finish in one. This should be repaired using products and techniques specific to EIFS.