We present an HI 21cm absorption study of a sample of 26 radio-loud active galactic nuclei (AGN) at $0.25 < z < 0.4$ carried out with the Karl G. Jansky Very Large Array. Our aim was to study the rate of incidence of HI in various classes of radio AGN, the morphology and kinematics of the HI, and the nature of the interaction between the HI and the radio source. Our sample consists of 14 extended sources and 12 compact sources in the radio-power range 10$^{25.7}$W/Hz$~-~10^{26.5}$W/Hz. We detect HI in 5 sources with a detection rate of $sim$19%, similar to the detection rate at lower redshifts. The rest-frame UV luminosities of most of the sources in the sample, including all the detections, are below the proposed threshold above which the HI is supposed to have been ionised. The optical emission-line spectra show that despite their high radio powers, one-third of the sample, including two detections, are low-ionisation sources. The radio continuum emission from the HI detections is unresolved at kpc scales, but is extended on parsec scales. The detections have complex HI 21cm absorption profiles with FWZI ranging from 60 km/s to 700 km/s and exhibit remarkably high HI column densities in the range 10$^{21}$ cm$^{-2}$ to 10$^{22}$ cm$^{-2}$ for T$_{rm spin}=$100 K and unit covering factor. A modelling of the HI 21cm absorption profiles suggests that in 2 sources the gas is disturbed, and in 3 cases, including the one with disturbed HI, the majority of the absorption is consistent with arising from an HI disc. Though we detect no fast HI outflows, the optical emission lines in the HI detections show the presence of highly disturbed gas in the nuclear region. Since some of our detections are low-ionisation AGN, this disturbance may be caused by the radio jets. Overall, our findings point towards a continuation of the low-$z$ trends in the incidence of HI in radio AGN up to $z sim 0.4$.
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