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According to Genki II, "any causative verb can be interpreted either with the "make somebody do" reading or the "let somebody do" reading".
In English, it's not uncommon to hear a construction such as "Let it drop" or "Make him do it". So are there constructions such as 「落ちさせて」?
I do not know what “causative verb” means in your book, but judging from the 落ちさせる example, I assume that it means verbs with -せる/させる suffix such as 遊ばせる and 片付けさせる. (使役動詞 in Japanese usually means a different form: it means verbs such as 落ちさす, 遊ばす, and 片付けさす.)
Then you can just use them in the imperative form with -て. A child may say to their parents:
宿題が終わったらテレビゲームで遊ばせて。 Let me play a video game after I finish homework.
If a father always cleans his child’s room, a mother may say to him:
遥に自分で片付けさせて。 Make Haruka clean up her own room.
“Let it drop.” in English would be 落ちさせて in Japanese if we only considered grammar, but we just do not say it that way. If “let it drop” means something like “do not worry about it,” it is usually just 気にしないで or 放っておいて.
