Despite highly developed specific immune responses, tumour cells often manage to escape recognition by the immune system, continuing to grow uncontrollably. Experimental work suggests that mature myeloid cells may be central to the activation of the specific immune response. Recognition and subsequent control of tumour growth by the cells of the specific immune response depend on the balance between immature (ImC) and mature (MmC) myeloid cells in the body. However, tumour cells produce cytokines that inhibit ImC maturation, altering the balance between ImC and MmC. Hence, the focus of this manuscript is on the study of the potential role of this inhibiting mechanism on tumour growth dynamics. A conceptual predator-prey type model that incorporates the dynamics and interactions of tumour cells, CD8(+) T cells, ImC and MmC is proposed in order to address the role of this mechanism. The prey (tumour) has a defence mechanism (blocking the maturation of ImC) that prevents the predator (immune system) from recognizing it. The model, a four-dimensional nonlinear system of ordinary differential equations, is reduced to a two-dimensional system using time-scale arguments that are tied to the maturation rate of ImC. Analysis shows that the model is capable of supporting biologically reasonable patterns of behaviour depending on the initial conditions. A range of parameters, where healing without external influences can occur, is identified both qualitatively and quantitatively.