  Absolute cross sections for the {\it K}-shell photoionisation of Be-like
(O$^{4+}$) and Li-like (O$^{5+}$) atomic oxygen ions were measured for the
first time (in their respective {\it K}-shell regions) by employing the
ion-photon merged-beam technique at the SOLEIL synchrotron-radiation facility
in Saint-Aubin, France. High-resolution spectroscopy with E/$\Delta$E $\approx$
3200 ($\approx$ 170 meV, FWHM)was achieved with photon energy from 550 eV up to
670 eV. Rich resonance structure observed in the experimental spectra is
analysed using the R-matrix with pseudo-states (RMPS) method. Results are also
compared with the screening constant by unit nuclear charge (SCUNC)
calculations. We characterise and identify the strong $\rm 1s \rightarrow 2p$
resonances for both ions and the weaker $\rm 1s \rightarrow np$ resonances ($ n
\ge 3$) observed in the {\it K}-shell spectra of O$^{4+}$.
