In Japanese, なくて is a construction that combines ない (in its conjugation form, なく), with the 接続助詞（せつぞくじょし） て. This results in the same ‘and’ meaning as the standard て form, but with the negative meaning of ない.
なくて is often used when the phrase that comes before it is being highlighted as an example/reason for a specific result. In these situations, the translation is usually ‘because not (A), (B)’, or ‘because I didn’t (A), (B)’.

• あなたが来（こ）れなくて残念（ざんねん）です。
  It is unfortunate that you were not able to come. (Not being able to come is the reason that it was unfortunate)
• 今日（きょう）は寒（さむ）くはなくて暖（あたた）かいから、今日（きょう）はジャケットを着（き）なくてもいいね。
  Because it is not cold today, but warm, we don't have to wear our jackets, right? (Not being cold is the reason that it was hot)
• このスマホは便利（べんり）ではなくて残念（ざんねん）だ。
  Since this smartphone is not useful, it’s a letdown.
• 虫歯（むしば）ではなくて安心（あんしん）した。
  I am relieved because it was not a cavity.

When なくて is used in this way, it is often followed by the speaker's feelings (in the form of an adjective), verbs in the potential form, or something that is generally beyond the control of the speaker. It is due to this fact that the result is ‘beyond control’ that the translation becomes ‘because’.

• 昨日（きのう）は良（よ）く寝（ね）れなくて、疲（つか）れている。
  I am tired because I did not sleep well yesterday.
• 明日（あした）は仕事（しごと）がなくて嬉（うれ）しい。
  I am happy because I don't have work tomorrow.

Some examples of words that express things beyond the control of the speaker are 困（こま）る (to be troubled), 嬉（うれ）しい (to be happy), 疲（つか）れている (to feel weary), 心配（しんぱい） (to feel anxiety/worry), 安心（あんしん） (to feel relief).