boolean circuit to decide if there's a path of at most $k$ edges from $u$ to $v$ in graph $G$
Since you are allowed an $n^{O(k)}$ size circuit, you could go over all possible $k$-length paths originating at $u$. The first level will contain $n^{O(k)}$ AND gates, one for every possible sequence of vertices $v_1=u,v_2,...,v_l=v$, for $l\le k$, and in the second level you input the results of the previous level to an OR gate.