Let $W_t$ denote the continuous time Wiener process at time $t$. Then its time derivative $\xi(t):=\frac{d W_t}{dt}$ is white noise, and may be "seen" in the weak sense via inner products $\langle \xi, \psi\rangle = \langle W', \psi\rangle = \int \psi dW$ where $\psi$ is a test function.
From this SO question, I learned that we can make similar statements about the second derivative of the Wiener process, or the derivative of white noise. Namely,
$$ \langle W'', \psi\rangle = \langle W, \psi''\rangle = \int \psi''dW $$
I am interested in a related quantity, namely $\langle (W'')^2, \psi\rangle $. Is there some way to express this random variable as an integral of (some function of) $\psi$ with respect to the Wiener process?
