A personal computer is something kids will use study at home

A personal computer is something that kids will use study at home(定语从句）

A personal computer is something kids will use to study at home.

即use后加to,因为两个动词use和 study不能直接连用。

A personal computer is something that kids will use to study at home.
that后是定语从句，不定代词something做先行词，that不能省略，to study at home做目的状语。

A personal computer is something kids will use-for-study at home

技术不行，仅供参考