In this paper, we consider three hitting problems on a disk intersection graph: Triangle Hitting Set, Feedback Vertex Set, and Odd Cycle Transversal. Given a disk intersection graph $G$, our goal is to compute a set of vertices hitting all triangles, all cycles, or all odd cycles, respectively. Our algorithms run in time $2^{\tilde O(k^{4/5})}n^{O(1)}$, $2^{\tilde O(k^{9/10})}n^{O(1)}$, and $2^{\tilde O(k^{19/20})}n^{O(1)}$, respectively, where $n$ denotes the number of vertices of $G$. These do not require a geometric representation of a disk graph. If a geometric representation of a disk graph is given as input, we can solve these problems more efficiently. In this way, we improve the algorithms for those three problem by Lokshtanov et al. [SODA 2022].