We present a spatial hybrid discrete-continuum modelling framework for the interaction dynamics between tumour cells and cytotoxic T cells, which play a pivotal role in the immune response against tumours. In this framework, a stochastic individual-based model for the dynamics of tumour cells and T cells is coupled with a balance equation for the concentration of chemokines that drive the chemotactic movement of T cells towards the tumour. We formally derive the deterministic continuum counterpart of this model, which is given by a coupled system that comprises an integro-differential equation for the density of tumour cells, a partial differential equation for the density of T cells, and a partial differential equation for the concentration of chemokines. We report on computational results of the hybrid model and show that there is an excellent quantitative agreement between them and numerical solutions of the corresponding continuum model. These results shed light on the mechanisms that underlie the emergence of different levels of infiltration of T cells into the tumour and elucidate how T-cell infiltration shapes anti-tumour immune response. Moreover, exploiting the computational efficiency of the continuum model, we carry out extensive numerical simulations to investigate the impact of T-cell infiltration on the response of tumour cells to different types of anti-cancer immunotherapy.