Context. The origins of quiet-Sun magnetism (QS) is still under debate and investigating the solar cycle variation observationally in greater detail can provide clues on how to resolve the ensuing controversies. Aims. We investigate the solar cycle variation of the most magnetically quiet regions and their surface gravity oscillation (f-) mode-integrated energy, E-f. Methods. We used 12 years of Helioseismic and Magnetic Imager (HMI) data and applied a stringent selection criteria based on spatial and temporal quietness to avoid any influence from active regions (ARs). We developed an automated high-throughput pipeline to go through all available magnetogram data and to compute the value of E-f for the selected quiet regions. Results. We observed a clear solar cycle dependence of the magnetic field strength in the most quiet regions containing several supergranular cells. For patch sizes smaller than a supergranular cell, no significant cycle dependence was detected. The E-f at the supergranular scale is not constant over time. During the late ascending phase of Cycle 24 (SC24, 2011-2012), it is roughly constant, but starts diminishing in 2013, as the maximum of SC24 is approached. This trend continues until mid-2017, when hints of strengthening at higher southern latitudes are seen. Slow strengthening continues, stronger at higher latitudes than at the equatorial regions, but E-f never returns to the values seen in 2011-2012. In addition, the strengthening trend continues past the solar minimum, to the years when SC25 is already clearly ascending. Hence, the E-f behavior is not in phase with the solar cycle. Conclusions. The dependence of E-f on the solar cycle at supergranular scales is indicative of the fluctuating magnetic field being replenished by tangling from the large-scale magnetic field - and not solely due to the action of a fluctuation dynamo process in the surface regions. The absence of variations on smaller scales might be an effect of the limited spatial resolution and magnetic sensitivity of HMI. The anticorrelation of E-f with the solar cycle in gross terms is expected, but the phase shift of several years indicates a connection to the large-scale poloidal magnetic field component rather than the toroidal one. Calibrating AR signals with the QS E-f does not reveal significant enhancement of the f-mode prior to AR emergence.