Vehicle platoons (VPs) are groups of vehicles driving together with a short inter-vehicle gap and a harmonized velocity. For a long period, the VPs and human-driven vehicles (HDVs) will coexist in mixed traffic flow, where the cut-in maneuver of the HDVs towards the VPs can be frequently expected. In this paper, to handle such cut-ins, we propose an intention prediction-based control method for the VPs by considering the tradeoff between the platoon integrity and traffic safety. Particularly, the proposed method is designed to prevent as many cut-ins as possible while taking care of the road safety. It consists of a cut-in prediction part, including intention and trajectory prediction algorithms, and a finite state machine (FSM)-based predictive control part, including a high-level FSM and a low-level predictive control. Driver-in-the-loop experiments were conducted in the VP-based driving scenarios to train the intention prediction algorithm and test the proposed method. We show the results detailing the control behavior of the proposed method in a no cut-in test, a mandatory cut-in test, and three discretionary cut-in tests. The results demonstrate that the proposed method can predict the cut-in intention of human drivers in real time. Besides, according to the prediction results, the proposed method can prevent cut-ins for the VPs while taking care of the road safety.