Unless you mean something else when you write $\psi$ class, it is expressible in terms of boundary divisors.

That is, if $\psi_i$ is the $i$-th cotangent bundle, then you can write it in terms of boundary divisors. One reference for this is the tome "Mirror Symmetry" by Hori, Katz, Klemm, et al. on p. 513, the comparison lemma.

The idea is that you can consider the forgetful maps $\pi$ from $M_{0,n}$ to $M_{0,n-1}$ and look at the divisor
$$
\psi_i - \pi^*\psi_i
$$
(where the $\psi$ classes are, by abuse of notation, living on different spaces). This is expressible in terms of boundary divisors, and so we can inductively write out the $\psi$ classes on any $M_{0,n}$.