How to select p terms with powers larger than 1 in a given expression?

My expression is:

$ expr = \frac{p^{\frac{1}{3}} q + 5}{3q^2} + p^2 + (p -1)^{\frac{1}{3}}$

And I want to single out all terms with $ p^k, k > 1$

How should I do that? (If there is any way other than applying Series[] would be the best, since I want it to be less time-consuming when dealing with large expr.)