  We compute the eigenvalue fluctuations of uniformly distributed random
biregular bipartite graphs with fixed and growing degrees for a large class of
analytic functions. As a key step in the proof, we obtain a total variation
distance bound for the Poisson approximation of the number of cycles and
cyclically non-backtracking walks in random biregular bipartite graphs, which
might be of independent interest. We also prove a semicircle law for random
$(d_1,d_2)$-biregular bipartite graphs when $\frac{d_1}{d_2}\to\infty$. As an
application, we translate the results to adjacency matrices of uniformly
distributed random regular hypergraphs.
