More precise form of the central limit theorem for $U$-statistics of $m$-dependent variables

Let $X_1,X_2,\dots$ be a stationary sequence of $m$-dependent random variables and let $\Phi(x_1,\dots,x_r)$ be a symmetric function. For the distribution of the $U$-statistics
$$ U_n=(C_n^r)^{-1}\sum_{1\le i_1<\dots<i_r\le n}\Phi(X_{i_1},\dots,X_{i_r}) $$
the rate of convergence to the normal law is investigated.