Flow-induced vibration (FIV) of a circular cylinder with an attached splitter plate in a laminar flow with Re = 100 is studied numerically. First, the mechanical model along with mathematical formulations is proposed to describe the fluid-structure interaction (FSI) between the elastically supported cylinder–plate body and the surrounding flow. Subsequently, an FSI solution procedure is developed based on the characteristic-based split finite element method, and its accuracy and stability are validated using vortex-induced vibrations (VIVs) of a plain circular cylinder with benchmark solutions. Finally, using FSI simulations, effects of the plate length (L), reduced velocity, mass ratio and damping coefficient on the dynamic response, fluid load, and flow pattern of the cylinder–plate assembly are investigated in detail. As the plate length increases from L/D = 0–1.5 (D is the cylinder diameter), three FIV modes are observed successively: VIV, coupled VIV and galloping, and separated VIV and galloping, along with three vortex modes in the wake: 2S (two separated vortices in one cycle), P+S (a vortex pair and a separated vortex in one cycle), and 2P (two vortex pairs in one cycle). Moreover, it is found that the lift components generated from the splitter plate and the cylinder behave, respectively, as the driving force and the suppressing force of galloping, and the transition from VIV to galloping can be taken as a result of the competition between them. The cylinder–plate model presented could be taken as a benchmark model demonstrating the VIV-galloping interaction and applied to the design of novel FSI-based energy harvesters.