Translation:The enemy has made a mistake.
I am slightly confused by -pu', contrasted with its absence. In this example, what would this sentence mean, practically, with and without the suffix? In English, "has made a mistake" seems to indicate that it has happened at some unspecified point in the past, while (correct me if I'm wrong) Qaghpu' is supposed to refer to a specific event of mistake-making in the (recent?) past.
I don't think there's any disagreement or controversy about using the Klingon perfective aspect for the English perfect tenses. You are correct in that Qaghpu' refers to a specific completed mistake-making, but the completed mistake-making is not limited to the (recent) past and could still be in the past, present, or future. "Has made a mistake" also refers to a specific completed mistake-making. To expand the tenses (since this Klingon sentence does not indicate tense) Qaghpu' could also be translated as "had made a mistake" and "will have made a mistake."
Where the mismatch of Klingon perfective and English perfect causes problems is for the English simple past. The English simple past is often also used to refer to specific completed events and so it is often appropriate to match an English simple past tense with the Klingon perfective tense. So for instance, depending on context, you could translate the above Klingon sentence as, "The enemy made a mistake."
Without the -pu' suffix, this sentence would not make any reference to the mistake-making being complete and would likely refer to a habitual, non-specific, or not yet started action. Some believe that the restriction is very strong others believe there are some overlapping edges. There are some areas where you might use the simple past to refer to habitual or non-specific actions and so you would use the verb without -pu' for those actions, even in the past. This course requires you to leave -pu' off when matching the English simple past, but that is not without controversy.
I hope that helped.
I think I understand. Klingon perfective aspect makes the verb refer to a single, specific instance (or maybe set of instances) of something happening, while imperfective may refer to a habit or something that is not specifically completed. As an English speaker, and not a linguist by any means, I suppose I kind of take aspect for granted. Kind of makes aspect suffixes in Klingon some of the most interesting, if not the most difficult to understand.
Klingon verbs do not indicate tense. The suffix -pu' doesn't put the sentence in the past; it just says the action is completed. It might be, for instance, completed in the future, and you're thinking of a time in which that is true.
wa'leS Qaghpu' jagh
Tomorrow the enemy will have made a mistake.
If you don't have the -pu' (or -ta') then the action is not completed (and without a -taH or -lI' the action is also not continuous).
The enemy makes a mistake.
This might be used in a sentence like qaStaHvIS Hoch may' Qagh jagh During every battle, the enemy makes a mistake. Or you might use it in a subordinate clause, like Qaghchugh jagh yIHIv If the enemy makes a mistake, attack! These sorts of meanings, where the action is neither completed nor continuous, use no aspect suffix. But they still don't tell you when something happens; that's tense, and Klingon doesn't have it in its verbs.