  Context. Very high-energy (VHE) $\gamma$-ray measurements of distant TeV
blazars can be nicely explained by TeV spectra induced by ultra high-energy
cosmic rays.
  Aims. We develop a model for a plausible origin of hard spectra in distant
TeV blazars. Methods. In the model, the TeV emission in distant TeV blazars is
dominated by two mixed components. The first is the internal component with the
photon energy around 1 TeV produced by inverse Compton scattering of the
relativistic electrons on the synchrotron photons (SSC) with a correction for
extragalactic background light absorbtion and the other is the external
component with the photon energy more than 1 TeV produced by the cascade
emission from high-energy protons propagating through intergalactic space.
  Results. Assuming suitable model parameters, we apply the model to observed
spectra of distant TeV blazars of 1ES 0229+200. Our results show that 1) the
observed spectrum properties of 1ES 0229+200, especially the TeV $\gamma$-ray
tail of the observed spectra, could be reproduced in our model and 2) an
expected TeV $\gamma$-ray spectrum with photon energy $>$1 TeV of 1ES 0229+200
should be comparable with the 50-hour sensitivity goal of the Cherenkov
Telescope Array (CTA) and the differential sensitivity curve for the one-year
observation with the Large High Altitude Air Shower Observatory (LHAASO).
  Conclusions. We argue that strong evidence for the Bethe-Heitler cascades
along the line of sight as a plausible origin of hard spectra in distant TeV
blazars could be obtained from VHE observations with CTA, LHAASO, HAWC, and
HiSCORE.
