Y-DNA would not help your search, contrary to Marion's suggestion. There are a few reasons for this. You would need to test at least 2 males to check your theory and that would mean first finding another descendant or male relative of the "secret lover" who is willing to test. Additionally, Y-DNA is relatively expensive ($129 to $169 at Family Tree DNA for the cheaper tests) and these tests' results are a bit harder to interpret.
You are better off doing an autosomal test -- 23andMe, Ancestry DNA, or MyHeritage. Most are on sale right now; and even at regular prices they can be considerably cheaper. With an autosomal test, you are not limited to males - the sex really doesn't matter. In your case, it's probably a good idea to test two or three people still, as that could help firm up any ambiguity. There's a bonus with this as well, since you won't just have your test takers to match against, but hundreds to thousands of others in these companies' databases. Just make sure that you and your relatives that test are all testing with the same company. Ancestry is a bit bigger (15+ million users) compared with 23andMe (10+ million users) and MyHeritage (2.5+ million users), however 23andMe and MyHeritage have better tools for checking relationships (a "chromosome browser") that would be specifically helpful to your case. Additionally, if you test males with 23andMe, it will also give you some info on whether they have the same Y-chromosome.
For your case, you should test yourself and/or one (or more) of your aunts/uncles on that side who are also descended from your grandfather, as well as one (or more) of your grandfather's siblings (your great aunts/uncles) if some are descended from the same father. It will be possible, from the amount of DNA that each of you share, to determine if there's a full-sibling or a half-sibling relationship between your grandfather and his siblings. The people who are closest to the uncertain relationship are the best to ones to test.
There are good statistics on how much DNA should be shared: https://dnapainter.com/tools & https://dnapainter.com/tools/sharedcm
Additionally, from these tests, you will have other DNA matches, some of which will be related through your grandfather's biological father. I'm not sure how common DNA testing for genealogy is in South Africa, but this might be a long shot... but after that it becomes a matter of patience. Try to figure out which test is most commonly used in your neck of the woods, since that will likely give you the best shot. It doesn't hurt to test with multiple companies (besides hurting one's pocketbook), so if you're able, that may also be worthwhile.