关於奥巴马写给她女儿们的一封信里一句话的语法问题?

这句话没有语法错误.
翻译： 但是我也知道那对你们和妈妈来说是不容易的（其中“那”可能指代的是这句话前面的内容,这里LZ没贴出来）,我还知道你们对那只新来的小狗感到有多么地兴奋,但是这些都不能弥补那些我们分别的时光.
分析：
1.But I also know that it hasn't always been easy for you and Mom
答：“I ”是主语, “Know”是谓语,“it hasn't always been easy for you and Mom”宾语从句.
2.and that as excited as you both are about that new puppy
答：这句话其实是与前面一句话并列的,由“and”连接. 其实两句话共享一个“主语+谓语”,即“I also know”. 这里涉及到两个词组：“As...as”和“be exited about something” 所以这句话只是把“exited”放在了 AS AS 的中间,后面拖一个ABOUT. “PUPPY”可以解释为小狗,幼犬.
3.it doesn't make up for all the time we've been apart.
答： 这里涉及到一个词组："make up for"解释为弥补. “all the time”作宾语,而后面的“we've been apart”是作定于从句修饰“all the time”的.
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希望能帮到你.

But I also know that (it hasn't always been easy for you and Mom), and that as excited as you both are (about that new puppy), it doesn't make up for all the time we've been apart.

但我也知道这并不总是容易为你和妈妈，而且一样兴奋，你都是对新的小狗，它不会为所有我们去过的时光。