Core prominence as a signature of restarted jet activity in the LOFAR radio-galaxy population

(abridged) Characterizing duty cycles of recurrent phases of dormancy and activity in supermassive black holes in active galactic nuclei is crucial in understanding impact of energy released on host galaxies and their evolution. However, identifying sources in quiescent and restarted phases is challenging. Our goal is to identify and characterize a substantial sample of radio galaxies in restarted phase and explore core prominence as a signature of this activity. We expand our prior study from a $30\,\mathrm{deg^2}$ area in Lockman Hole to a larger $424\,\mathrm{deg^2}$ region in Hobby-Eberly Telescope Dark Energy Experiment (HETDEX) field using visually selected core-dominated radio galaxies. We used 144 MHz LOFAR survey images to identify galaxies with restarting jets. By applying selection criteria, including radio core dominance, low surface brightness extended emission, spectral index properties, and morphology, we found 69 candidate restarted radio galaxies. These candidates show diverse intrinsic morphology, spanning FRI, FRII, core-with-halo, and asymmetric forms, suggesting different progenitors. Among these, nine galaxies exhibit ultra-steep spectrum extended emission combined with high radio core prominence, representing previous and current epochs of jet activity. This subset supports a model where the switch-on and switch-off mechanism occurs with fast duty cycle. The restarted candidates span radio luminosities from log$_{10}$(L$_\mathrm{144 MHz}$/$\mathrm{WHz^{-1}}$) = 23.24 to 26.80, with sizes between 88 and 1659 kpc, including 16 giant radio galaxies. Their total stellar content aligns with massive elliptical galaxies. Our findings at $z