The fundamental plane of black hole activity describes the correlation between radio luminosity (L_R_), X-ray luminosity (L_X_), and black hole mass (M_BH_). It reflects a connection between the accretion disc and the jet. However, the dependence of the fundamental plane on various physical properties of active galactic nuclei (AGNs) and host galaxies remains unclear, especially for low-luminosity AGNs, which is important for understanding the accretion physics in AGNs. Here, we explore the dependence of the fundamental plane on the radio loudness, Eddington-ratio (lambda_Edd_), redshift, and galaxy star formation properties (star-forming galaxies and quiescent galaxies) across 0.1<z<=4 for radio AGNs. Based on current deep and large surveys, our studies can extend to lower luminosities and higher redshifts. From the deep and large multi-wavelength surveys in the GOODS-N, GOODS-S, and COSMOS/UltraVISTA fields, we constructed a large and homogeneous radio AGN sample consisting of 208 objects with available estimates for L_R_ and L_X_. Then we divided the radio AGN sample into 141 radio-quiet AGNs and 67 radio-loud AGNs according to the radio loudness defined by the ratio of L_R_ to L_X_, and explored the dependence of the fundamental plane on different physical properties of the two populations, separately. The ratio of L_R_ to L_X_ shows a bimodal distribution that is well described by two single Gaussian models. The cross point between these two Gaussian components corresponds to a radio-loudness threshold of log (L_R_/L_X_)=-2.73. The radio-quiet AGNs have a significantly larger Eddington ratio than the radio-loud AGNs. Our radio-quiet and radio-loud AGNs show a significantly different fundamental plane, which indicates a significant dependence of the fundamental plane on the radio loudness. For both radio-quiet and radio-loud AGNs, the fundamental plane shows a significant dependence on lambda_Edd, but no dependence on redshift. The fundamental plane shows a significant dependence on the galaxy star formation properties for radio-quiet AGNs, while for radio-loud AGNs this dependence disappears. The fundamental plane sheds important light on the accretion physics and X-ray emission origins of central engines. X-ray emission of radio-quiet AGNs at 0.01<lambda_Edd_<0.1 are produced by a combination of advection-dominated accretion flow (ADAF) and synchrotron radiation from the jet, while at 0.1<lambda_Edd_<1 they mainly follow the synchrotron jet model. The origins of X-ray emission of radio-loud AGNs are consistent with a combination of ADAF and the synchrotron jet model at lambda_Edd_<0.01, agree with the synchrotron jet model at 0.01<lambda_Edd_<0.1, and follow a combination of the standard thin disc and a jet model at lambda_Edd_>0.1.