I came across this sentence
来たばかりの人間を、普通に呼び出さないで欲しい
Which I believe means something like
I want you not to so naturally summon someone who's just arrived
Please correct me if I'm off.
What I don't understand here is the usage of ないで欲しい instead of なくて, which is the negative te form. Are these 2 interchangeable? can you conjugate ないで with other te form constructions?
First, we need to establish that both なくて and ないで are the proper te-forms of negative verbs. We know that negative verbs function just like i-adjectives, and that it mostly true,
良い[adj] -> 良く[adv] -> 良ければ[if adj], etc
話さない -> 話さなく -> 話さなければ, etc.
except for one thing: while true i-adjectives have only one te-form ーくて ( 良くて、悪くて、早くて、清くて), negative verbs have two te-form ーくて and ーいで (話さなくて・話さないで、見なくて・見ないで、行かなくて・行かないで)
Instead of ーくて and ーいで we can remember them as ーなくて and ーないで since negative verbs always end in ない, and the い changes to くて or いで.
The difference? We know that te-form is used to connect clauses, and when the previous clause expresses reason, or that the latter clause defines or describes the previous clause, generally なくて is used:
今日は車が発動できなくてバスで来ました。[Reason] The first clause explains the action in the second
雨が降らなくて良かった [Description] 良かった describes the fact that 雨が降らない
見なくてもいいです [Description] いい describes the action of 見ない
When dealing with state-of-being, ないで is preferred:
砂糖を入れないでコーヒーを飲む. [State] The state of the coffee is without sugar.
取扱説明書を読まないで使う [State] To use it in the state of not having read the manual
Note that for describing a state, the adverbial form of the negative verbs also work:
砂糖を入れなくコーヒーを飲む
取扱説明書を読まなく使う
Lastly, when it comes to imperatives, requests, wishes or demands, ONLY ないで is used. This directly relates to your question:
開けないでください！⭕️
開けなくてください！❌
開けないで！⭕️
開けなくて！❌
開けないで欲しいよ⭕️
開けなくて欲しいよ❌
開けないで欲しいよ⭕️
開けなくて欲しいよ❌
For imperatives, [I think] using なくて in lieu of ないで is plain-out grammatically wrong, but for the others usages above なくて and ないで are still somewhat interchangeable to a certain degree. Someone who is more knowledgeable about the specific nuances of なくて and ないで are welcome to add to my answer.
