The second one is wrong!

This would be the correct sentence "She gave me what little food she had" she gave me little food she had => in this sentence you are missing the word "the" which is and article (article is a part of speech, like an adjective because it modifies a noun) or like the correct sentence above you can use the word "what". :)

There is no difference they both mean, she had given the last of her food to someone else