Non-LTE analysis (LTE is local thermodynamic equilibrium) of the oxygen abundances for 51 Galactic A-, F- and G-type supergiants and bright giants is performed. In contrast with carbon and nitrogen, oxygen does not show any significant systematic anomalies in their abundances log E(O). There is no marked difference from the initial oxygen abundance within errors of the log E(O) determination across the Teff interval from 4500 to 8500 K and the log g interval from 1.2 to 2.9 dex. This result agrees well with theoretical predictions for stellar models with rotation. With our new data for oxygen and our earlier non-LTE determinations of the N and C abundances for stars from the same sample, we constructed the [N/C] vs [N/O] relation for 17 stars. This relation is known to be a sensitive indicator of stellar evolution. A pronounced correlation between [N/C] vs [N/O] is found; the observed [N/C] increase from 0 to 1.6 dex is accompanied by the [N/O] increase from 0 to 0.9 dex. When comparing the observed [N/C] vs [N/O] relation with the theoretical one, we show that this relation reflects a strong dependence of the evolutionary changes in CNO abundances on the initial rotation velocities of stars. Given that the initial rotational velocities of these stars are expected to satisfy V0<150 km/s, it is found that they are mostly the post first dredge-up (post-FDU) objects. It is important that just such initial velocities V0 are typical for about 80% of stars in question (i.e. for stars with masses 4-19 M_sun). A constancy of the total C+N+O abundance during stellar evolution is confirmed. The mean value log E(C+N+O)=8.97+/-0.08 found for AFG supergiants and bright giants seems to be very close to the initial value 8.92 (the Sun) or 8.94 (the unevolved B-type MS stars).
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