  We consider a non-local tumour growth model of phase-field type, describing
the evolution of tumour cells through proliferation in presence of a nutrient.
The model consists of a coupled system, incorporating a non-local Cahn-Hilliard
equation for the tumour phase variable and a reaction-diffusion equation for
the nutrient. First, we establish novel regularity results for such a model, by
applying maximal regularity theory in weighted $L^p$ spaces. This technique
enables us to prove the local existence and uniqueness of a regular solution,
including also chemotaxis effects. By leveraging time-regularisation properties
and global boundedness estimates, we further extend the solution to a global
one. These results provide the foundation for addressing an optimal distributed
control problem, aimed at identifying a suitable therapy guiding the tumour
towards a predefined target. Specifically, we prove the existence of an optimal
therapy and, by studying the Fr\'echet-differentiability of the
control-to-state operator and introducing the adjoint system, we derive
first-order necessary optimality conditions.
