I'm hoping that this is an easy question for someone.

How many elements can we expect to have multiplicative order at most $n^{1/c}$ in one of the finite fields $\mathbb{F}_p$ with $p$ prime with $n \le p \le n^2$, for $c=\frac{9\log{n}}{\log{\log{\log{n}}}}$?

Note that I'm trying to bound the expectation from below.

CLARIFICATION

We're taking the average order in one field, out of the set of finite fields with $n \le p \le n^2$.