A hydrogen-bonding network or hydrogen-bonded cluster is formed in many hydrogen-bonded glass formers. It determines the dynamics of structural α relaxation and the Johari-Goldstein (JG) β relaxation because breaking of hydrogen bonds is the prerequisite. However, the networks and clusters can be substantially reduced or totally removed in the liquid state by high temperature accompanying the applied high pressure in experiments, and in the glassy state by hyperquenching the liquid under pressure. By confining the glass former in nanometer spaces, the extended network cannot form, and in addition the finite size effect limits the growth of the length scale of the α relaxation on lowering temperature. Any of these actions will modify the structure of the original hydrogen-bonded glass former, and also the intermolecular interaction governing the relaxation processes. Consequently the dynamics of the structural α relaxation and the JG β relaxation, as well as the relation between the two processes, are expected to change. An important advance in the study of the dynamics of glass-forming materials is the existence of the strong connection between the α relaxation and the JG β relaxation. In particular, the ratio of their relaxation times, t_{α}(T)/t_{β}(T), is quantitatively determined by the exponent of the Kohlrausch relaxation function of the α relaxation. This property is valid in hydrogen-bonded glass formers as well as in non-hydrogen-bonded glass formers. The interesting question is whether this property continues to hold after the hydrogen-bonded glass former has been modified by high temperature under high pressure, nanoconfinement, and hyperquenching under pressure. Remarkably, the answer is positive as concluded from the analyses of the data in several hydrogen-bonded glass formers reported in this paper. So far the main theoretical explanation of this property has been the coupling model.