The magnetic anisotropy in MgO-capped MnPt films and its voltage control are studied using first-principles calculations. Sharp variation of the magnetic anisotropy with film thickness, especially in the Pt-terminated film, suggests that it may be widely tuned by adjusting the film thickness. In thick films the linear voltage control coefficient is as large as 1.5 and $-0.6$ pJ/Vm for Pt-terminated and Mn-terminated interfaces, respectively. The combination of a widely tunable magnetic anisotropy energy and a large voltage-control coefficient suggest that MgO-capped MnPt films can serve as a versatile platform for magnetic memory and antiferromagnonic applications.