The dynamics of the moving-with-constant-velocity internal pressure acting on the inner surface of the hollow circular cylinder surrounded by an infinite elastic medium is studied within the scope of the piecewise homogeneous body model by employing the exact field equations of the linear theory of elastodynamics. It is assumed that the internal pressure is point-located with respect to the cylinder axis and is axisymmetric in the circumferential direction. Moreover, it is assumed that shear-spring type imperfect contact conditions on the interface between the cylinder and surrounding elastic medium are satisfied. The focus is on the influence of the mentioned imperfectness on the critical velocity of the moving load and this is the main contribution and difference of the present paper the related other ones. The other difference of the present work from the related other ones is the study of the response of the interface stresses to the load moving velocity, distribution of these stresses with respect to the axial coordinates and to the time. At the same time, the present work contains detail analyses of the influence of problem parameters such as the ratio of modulus of elasticity, the ratio of the cylinder thickness to the cylinder radius, and the shear-spring type parameter which characterizes the degree of the contact imperfection on the values of the critical velocity and stress distribution. Corresponding numerical results are presented and discussed. In particular, it is established that the values of the critical velocity of the moving pressure decrease with the external radius of the cylinder under constant thickness of that.