We study an approximate controllability problem for the Cahn-Hilliard equation with terminal time observations at a finite number of spatial points. More precisely, for fixed time $T$ and a given $\varepsilon>0$, we investigate whether the solution of the system reaches the $\varepsilon$-neighbourhood of the desired state at predetermined finitely many points. In the first problem, we consider the admissible control set to be $L^2$. As a first step, we prove approximate controllability of the associated linearised system, through a variational approach, by introducing a suitable convex functional involving solutions of the adjoint equation with Dirac-type terminal data. The result is then extended to the semilinear system by using a fixed-point argument.\\

Moreover, we study the controllability problem for control acting at finitely many spatial points along with pointwise observations. In this setting, the pointwise evaluation of the state requires additional regularity of the solution; hence, we prove the result only for the linear model. To tackle finite pointwise control, we write the pointwise control as the limit of specific $L^2$ controls and use previously obtained results. Further, by employing compactness arguments, we conclude the approximate controllability.