I’m not sure whether this is correct in the exercise:
It says 규 is “kyu” but 교 is “gyo”. It should be both “g”, right? At the start of a syllable, the letter is always pronounced as a “g” as far as I know.
As I understand it, that Korean letter is right at the cusp of g/k, and isn’t ever really solidly a g or a k sound as in English. I’m not sure if the exercise you’re talking about requires you to type in English and isn’t letting you enter the answers with g or k swapped, but you do seem to be posting about a specific course, so you really ought to paste in a link to the course you’re posting about.
@cos is right. The ㄱ is somewhere between the English G and K. Sometimes it sounds a bit more like one or the other, depending on what surrounds it. I’ve noticed that it will usually sound more like a K if it’s at the beginning of a word. When romanizing Hangeul, they will usually just jump between using G or K, and it’s never really wrong either way. For example, you may have heard of the Korean city named Gwangju, which is often also called Kwangju.
All in all, that character represents a sound that does not exist in English. Using G and K are just a best effort at making Hangeul accessible to non-koreans. This tool here will help exercise your listening skills in this http://mykoreancoach.educasiainc.com/#pageTrainingType
It’s worth mentioning that Korean uses a different writing system for a reason. If the English Alphabet described their language well in written form, they would definitely be using it. But they don’t. Trying to learn Korean using the English Alphabet the way you’re doing will confine you to learning a version of Korean that is bound by the limitations of the English language. I highly recommend that you find a course that doesn’t prompt you to use the English Alphabet at all. Instead, buy a korean keyboard (or a korean overlay for your laptop keyboard) and find a course that asks you to type Hangeul. It is a bit painful at first but it will really pay off in the long run. I remember that How To Study Korean put out a good few courses that I followed when I was learning Korean.
Buying a Korean keyboard is really unnecessary. It can easily be installed on your computer, for free. All you have to do, is memorize the placements, but that doesn’t take long. Took me about two days to get used to it, and after that the typing speed was almost on the level of the roman keyboard.