Unfortunately, the premise of your question is wrong.
There is no cyclopropylcarbinyl or cyclobutyl cation. They're the same thing. Unlike cations which you might be more familiar with, for example, t-butyl cation, the cation you're looking at here, also called homoallyl, is a non-classical cation.
The open form and the two structures you drew in the second line are not distinct. They're just resonance forms of each other.
You ask does this ion undergo ring expansion? Well, yes and no. Yes, in that there is a contribution from the 4-member ring to the resonance structure. No, in that you can't expand something that doesn't exist.