Based on our prior accurate determination of fundamental parameters for 36 Galactic A- , F- and G-type supergiants and bright giants ( luminosity classes I and II ) , we undertook a non-LTE analysis of the carbon abundance in their atmospheres . It is shown that the non-LTE corrections to the C abundances derived from C I lines are negative and increase with the effective temperature T _ { eff } ; the corrections are especially significant for the infrared C I lines with wavelengths 9060-9660 Å . The carbon underabundance as a general property of the stars in question is confirmed ; a majority of the stars studied has the carbon deficiency [ C/Fe ] between -0.1 and -0.5 dex , with a minimum at -0.7 dex . When comparing the derived C deficiency with the N excess found by us for the same stars earlier , we obtain a pronounced N vs. C anti-correlation , which could be expected from predictions of the theory . We found that the ratio [ N/C ] spans mostly the range from 0.3 to 1.7 dex . Both these enhanced [ N/C ] values and the C and N anomalies themselves are an obvious evidence of the presence on a star ’ s surface of mixed material from stellar interiors ; so , a majority of programme stars passed through the deep mixing during the main sequence ( MS ) and/or the first dredge-up ( FD ) phase . Comparison with theoretical predictions including rotationally-induced mixing shows that the stars are either post-MS objects with the initial rotational velocities V _ { 0 } = 200-300 km/s or post-FD objects with V _ { 0 } = 0-300 km/s . The observed N vs. C anti-correlation reflects a dependence of the C and N anomalies on the V _ { 0 } value : on average the higher V _ { 0 } the greater the anomalies . It is shown that an absence of detectable lithium in the atmospheres of the stars , which is accompanied with the observed N excess and C deficiency , is quite explainable .