* little/ a little
- I was in bad mood. So i stayed for a little while in the shopping mall.
This sentence is good with a small correction: I was in a bad mood, so I stayed for a little while in the shopping mall.
- Please buy some milk when you get home. There is little milk in the refrigerator.
The sentence is good also. "Little" is less than "a little" or less than you'd like.
This is the same as saying: Please buy some milk when you get home. There is only a little milk left in the refrigerator.
Meaning you have some milk, but not enough.