What complexity class is this ciruit problem?
Assuming that the probability is over the randomized preprocessing and $p &gt; 0$, by fixing the correct random choice you get a circuit of size $S = O(2^{\log^{O(1)} n} q^{O(1)})$ for $k$-SAT. The corresponding complexity class is non-uniform circuits of size $S$. If, for example, $q$ is quasipolynomial ($2^{\log^{O(1)}n}$), the you get a non-uniform circuit of quasipolynomial size. 

We usually don't care too much about the time required to construct the circuit unless it is very low.