Confusion with the て form or verbs ending with く or ぐ

All the sources I have used told me the て form of verbs ending with く or ぐ should have that く or ぐ replaced with いて or いで, but often I actually see it being replaced with きて or きで instead, and I can’t find any explanation for that

For example I assume 泣く would become 泣いて and sometimes I see it like that, but I also see 泣きて sometimes. Another example is 生く to 生いて or 生きて

Which is correct? Or are both correct, and do they mean different things? Thanks for your help in advance