Catalyzes the excision of an oxidatively damaged form of guanine (7,8-dihydro-8-oxoguanine = 8-oxoG) from DNA. Also cleaves the DNA backbone at apurinic/apyrimidinic sites (AP sites). 2'-deoxyribonucleotide-(2'-deoxyribose 5'-phosphate)-2'-deoxyribonucleotide-DNA = a 3'-end 2'-deoxyribonucleotide-(2,3-dehydro-2,3-deoxyribose 5'-phosphate)-DNA + a 5'-end 5'-monophospho-2'-deoxyribonucleoside-DNA + H(+) Belongs to the type-2 OGG1 family.