Let $(X,\mathscr{B}, μ,T,d)$ be a measure-preserving dynamical system with exponentially mixing property, and let $μ$ be an Ahlfors $s$-regular probability measure. The dynamical covering problem concerns the set $E(x)$ of points which are covered by the orbits of $x\in X$ infinitely many times. We prove that the Hausdorff dimension of the intersection of $E(x)$ and any regular fractal $G$ equals $\dim_{\rm H}G+α-s$, where $α=\dim_{\rm H}E(x)$ $μ$--a.e. Moreover, we obtain the packing dimension of $E(x)\cap G$ and an estimate for $\dim_{\rm H}(E(x)\cap G)$ for any analytic set $G$.