Question

We'll do what we can ______ English well this term.

Answer 1

to study
动词不定式表目的。“这个学期我们将竭尽全力来学好英语。”
这是个省略句We’ll do what we can （do) to English well this term.当前面出现do 时can后面可以省略do . to English well this term.是目的状语，what we can （do)是宾语从句。其它类似的结构还有：
We’ll do what\ everything (that) \ all (that) + we can （do) to English well this term.

Answer 2

楼上正解，to learn，意为本学期我们将尽己所能学好英语。

Answer 3

do somthing well 固定短语

Answer 4

to learn/study

Answer 5

to learn