Do you have to use deep-fry oil or corn oil to make "African Drop Doughnuts" or can you use cooking oil?
It's hard to give a specific answer since I don't know what temperature you need. I looked at a couple recipes for African Drop Donuts and the temperatures were pretty far apart: 340F and 375F. I also don't know what you mean by cooking oil!
But you can use any oil suitable for deep-frying at the temperature specified in the recipe. Depending on what kind of cooking oil you have it may be fine, or it might have too low a smoke point.
Things like canola, sunflower, peanut, corn, and safflower would most likely be fine, since they all have smoke points of 400F or higher and are pretty neutral. You can look at a table of smoke points to get an idea beyond that.
The one thing you probably want to be careful about is oil labeled as generic vegetable oil; without knowing what kind of oil it actually is, it's hard to know what its smoke point is. It might also be more strongly flavored.