1. Can I go home a little early?
2. Can I go home kind of early?
3. Can I go home rather early?
4. Can I go home somewhat early?
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Are they all the same? Which ones are commonly used?
First, "can" implies an ability to do something. Anyone "can" go home early by merely doing so. To ask permission to leave early, one should ask, "MAY I go home...
Aside from that, all of these sentences are correct and they all mean the same thing. I prefer the first one. The others are either slangy (kind of) or rather stiff or formal (rather and somewhat).